Inflection Point and Concavity Intervals — Dynamic Practice

Inflection Point and Concavity Intervals — Dynamic Practice. Practice questions to deepen understanding of inflection points and concavity intervals. Online dynamic learning math system.

Dynamic practice in inflection points and concavity intervals — finding intervals where the function is concave up/down, identifying inflection points. New questions every attempt.

30 questions

Question 1

3.33 pts

Determine the concavity direction of the function at x = -3:

\(f(x) = -x^{3}+2x^{2}+2x+5\)

Concave up

Concave down

Inflection point

Cannot tell

Explanation:

Solution - Concavity at a Point

📌 The function:

\(f(x) = -x^{3}+2x^{2}+2x+5\)

📌 Step 1: First derivative

\(f'(x) = -3x^2+4x+2\)

📌 Step 2: Second derivative

\(f''(x) = -6x+4\)

📌 Step 3: Substitute x = -3 into the second derivative

\(f''(-3) = -6 \cdot -3 + 4\)

\(f''(-3) = 18 + 4\)

\(f''(-3) = 22\)

📌 Step 4: Check the sign

\(f''(-3) = 22\) > 0 (positive)

📌 Step 5: Conclusion

If f'(x) > 0	→	Concave up (∪)
If f'(x) < 0	→	Concave down (n)

Since \(f'(-3) > 0 (positive)\), the function is concave concave up (∪)

✓ Answer: Concave up

Question 2

3.33 pts

Determine the concavity direction of the function at x = 3:

\(f(x) = -3x^{3}+6x^{2}-2x-2\)

Concave down

Concave up

Inflection point

Cannot tell

Explanation:

Solution - Concavity at a Point

📌 The function:

\(f(x) = -3x^{3}+6x^{2}-2x-2\)

📌 Step 1: First derivative

\(f'(x) = -9x^2+12x-2\)

📌 Step 2: Second derivative

\(f''(x) = -18x+12\)

📌 Step 3: Substitute x = 3 into the second derivative

\(f''(3) = -18 \cdot 3 + 12\)

\(f''(3) = -54 + 12\)

\(f''(3) = -42\)

📌 Step 4: Check the sign

\(f''(3) = -42\) < 0 (negative)

📌 Step 5: Conclusion

If f'(x) > 0	→	Concave up (∪)
If f'(x) < 0	→	Concave down (n)

Since \(f'(3) < 0 (negative)\), the function is concave concave down (∩)

✓ Answer: Concave down

Question 3

3.33 pts

Determine the concavity direction of the function at x = 4:

\(f(x) = -x^{3}-2x^{2}-4x-4\)

Concave down

Concave up

Inflection point

Cannot tell

Explanation:

Solution - Concavity at a Point

📌 The function:

\(f(x) = -x^{3}-2x^{2}-4x-4\)

📌 Step 1: First derivative

\(f'(x) = -3x^2-4x-4\)

📌 Step 2: Second derivative

\(f''(x) = -6x-4\)

📌 Step 3: Substitute x = 4 into the second derivative

\(f''(4) = -6 \cdot 4 + -4\)

\(f''(4) = -24 + -4\)

\(f''(4) = -28\)

📌 Step 4: Check the sign

\(f''(4) = -28\) < 0 (negative)

📌 Step 5: Conclusion

If f'(x) > 0	→	Concave up (∪)
If f'(x) < 0	→	Concave down (n)

Since \(f'(4) < 0 (negative)\), the function is concave concave down (∩)

✓ Answer: Concave down

Question 4

3.33 pts

Determine the concavity direction of the function at x = 1:

\(f(x) = 2x^{3}-4x+3\)

Concave up

Concave down

Inflection point

Cannot tell

Explanation:

Solution - Concavity at a Point

📌 The function:

\(f(x) = 2x^{3}-4x+3\)

📌 Step 1: First derivative

\(f'(x) = 6x^2-4\)

📌 Step 2: Second derivative

\(f''(x) = 12x\)

📌 Step 3: Substitute x = 1 into the second derivative

\(f''(1) = 12 \cdot 1 + 0\)

\(f''(1) = 12 + 0\)

\(f''(1) = 12\)

📌 Step 4: Check the sign

\(f''(1) = 12\) > 0 (positive)

📌 Step 5: Conclusion

If f'(x) > 0	→	Concave up (∪)
If f'(x) < 0	→	Concave down (n)

Since \(f'(1) > 0 (positive)\), the function is concave concave up (∪)

✓ Answer: Concave up

Question 5

3.33 pts

Determine the concavity direction of the function at x = -2:

\(f(x) = 3x^{3}+4x^{2}+5x+5\)

Concave down

Concave up

Inflection point

Cannot tell

Explanation:

Solution - Concavity at a Point

📌 The function:

\(f(x) = 3x^{3}+4x^{2}+5x+5\)

📌 Step 1: First derivative

\(f'(x) = 9x^2+8x+5\)

📌 Step 2: Second derivative

\(f''(x) = 18x+8\)

📌 Step 3: Substitute x = -2 into the second derivative

\(f''(-2) = 18 \cdot -2 + 8\)

\(f''(-2) = -36 + 8\)

\(f''(-2) = -28\)

📌 Step 4: Check the sign

\(f''(-2) = -28\) < 0 (negative)

📌 Step 5: Conclusion

If f'(x) > 0	→	Concave up (∪)
If f'(x) < 0	→	Concave down (n)

Since \(f'(-2) < 0 (negative)\), the function is concave concave down (∩)

✓ Answer: Concave down

Question 6

3.33 pts

Find the inflection point of the function:

\(f(x) = -3x^{3}-3x^{2}-3x\)

\(x = \frac{-1}{3}\)

\(x = \frac{1}{3}\)

x = \(\frac{-1}{2}\)

No inflection point

Explanation:

Solution - Inflection Point

📌 The function:

\(f(x) = -3x^{3}-3x^{2}-3x\)

📌 Step 1: First derivative

\(f'(x) = -9x^2-6x-3\)

📌 Step 2: Second derivative

\(f''(x) = -18x-6\)

📌 Step 3: Solve f'(x) = 0

\(-18x-6 = 0\)

\(x = \frac{-1}{3}\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	\frac{-1}{3}	...	+∞
f''(x)		+	0	-
Concavity		∪	Inflection	∩

📌 Step 5: Conclusion

There is a sign change in the second derivative around x = \frac{-1}{3}

The concavity changes from ∪ to ∩

Therefore this is an inflection point.

✓ Answer: \(x = \frac{-1}{3}\)

Question 7

3.33 pts

Find the inflection point of the function:

\(f(x) = -2x^{3}-x^{2}+5x-2\)

\(x = \frac{-1}{6}\)

\(x = \frac{1}{6}\)

x = \(\frac{5}{2}\)

No inflection point

Explanation:

Solution - Inflection Point

📌 The function:

\(f(x) = -2x^{3}-x^{2}+5x-2\)

📌 Step 1: First derivative

\(f'(x) = -6x^2-2x+5\)

📌 Step 2: Second derivative

\(f''(x) = -12x-2\)

📌 Step 3: Solve f'(x) = 0

\(-12x-2 = 0\)

\(x = \frac{-1}{6}\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	\frac{-1}{6}	...	+∞
f''(x)		+	0	-
Concavity		∪	Inflection	∩

📌 Step 5: Conclusion

There is a sign change in the second derivative around x = \frac{-1}{6}

The concavity changes from ∪ to ∩

Therefore this is an inflection point.

✓ Answer: \(x = \frac{-1}{6}\)

Question 8

3.33 pts

Find the inflection point of the function:

\(f(x) = 2x^{3}+2x^{2}-4x-5\)

\(x = \frac{-1}{3}\)

\(x = \frac{1}{3}\)

x = 1

No inflection point

Explanation:

Solution - Inflection Point

📌 The function:

\(f(x) = 2x^{3}+2x^{2}-4x-5\)

📌 Step 1: First derivative

\(f'(x) = 6x^2+4x-4\)

📌 Step 2: Second derivative

\(f''(x) = 12x+4\)

📌 Step 3: Solve f'(x) = 0

\(12x+4 = 0\)

\(x = \frac{-1}{3}\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	\frac{-1}{3}	...	+∞
f''(x)		-	0	+
Concavity		∩	Inflection	∪

📌 Step 5: Conclusion

There is a sign change in the second derivative around x = \frac{-1}{3}

The concavity changes from ∩ to ∪

Therefore this is an inflection point.

✓ Answer: \(x = \frac{-1}{3}\)

Question 9

3.33 pts

Find the concavity intervals (concave up / concave down) of the function:

\(f(x) = 2x^{3}-4x^{2}-3x+2\)

Up: \((\frac{2}{3}, \infty)\), Down: \((-\infty, \frac{2}{3})\)

Up: \((-\infty, \frac{2}{3})\), Down: \((\frac{2}{3}, \infty)\)

Concave up everywhere

Concave down everywhere

Explanation:

Solution - Concavity Intervals

📌 The function:

\(f(x) = 2x^{3}-4x^{2}-3x+2\)

📌 Step 1: First derivative

\(f'(x) = 6x^2-8x-3\)

📌 Step 2: Second derivative

\(f''(x) = 12x-8\)

📌 Step 3: Find inflection points (f'(x) = 0)

\(12x-8 = 0\)

\(x = \frac{2}{3}\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	\frac{2}{3}	...	+∞
f''(x)		-	0	+
Concavity		∩	Inflection	∪

📌 Step 5: Conclusion

• Concave up (∪) on the interval: \((\frac{2}{3}, \infty)\)

• Concave down (n) on the interval: \((-\infty, \frac{2}{3})\)

✓ Answer: Up: \((\frac{2}{3}, \infty)\), Down: \((-\infty, \frac{2}{3})\)

Question 10

3.33 pts

Find the concavity intervals (concave up / concave down) of the function:

\(f(x) = -x^{3}-5x^{2}-2x\)

Up: \((-\infty, \frac{-5}{3})\), Down: \((\frac{-5}{3}, \infty)\)

Up: \((\frac{-5}{3}, \infty)\), Down: \((-\infty, \frac{-5}{3})\)

Concave up everywhere

Concave down everywhere

Explanation:

Solution - Concavity Intervals

📌 The function:

\(f(x) = -x^{3}-5x^{2}-2x\)

📌 Step 1: First derivative

\(f'(x) = -3x^2-10x-2\)

📌 Step 2: Second derivative

\(f''(x) = -6x-10\)

📌 Step 3: Find inflection points (f'(x) = 0)

\(-6x-10 = 0\)

\(x = \frac{-5}{3}\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	\frac{-5}{3}	...	+∞
f''(x)		+	0	-
Concavity		∪	Inflection	∩

📌 Step 5: Conclusion

• Concave up (∪) on the interval: \((-\infty, \frac{-5}{3})\)

• Concave down (n) on the interval: \((\frac{-5}{3}, \infty)\)

✓ Answer: Up: \((-\infty, \frac{-5}{3})\), Down: \((\frac{-5}{3}, \infty)\)

Question 11

3.33 pts

Determine the concavity direction of the function at x = -4:

\(f(x) = -3x^{3}-x^{2}-2x-5\)

Concave up

Concave down

Inflection point

Cannot tell

Explanation:

Solution - Concavity at a Point

📌 The function:

\(f(x) = -3x^{3}-x^{2}-2x-5\)

📌 Step 1: First derivative

\(f'(x) = -9x^2-2x-2\)

📌 Step 2: Second derivative

\(f''(x) = -18x-2\)

📌 Step 3: Substitute x = -4 into the second derivative

\(f''(-4) = -18 \cdot -4 + -2\)

\(f''(-4) = 72 + -2\)

\(f''(-4) = 70\)

📌 Step 4: Check the sign

\(f''(-4) = 70\) > 0 (positive)

📌 Step 5: Conclusion

If f'(x) > 0	→	Concave up (∪)
If f'(x) < 0	→	Concave down (n)

Since \(f'(-4) > 0 (positive)\), the function is concave concave up (∪)

✓ Answer: Concave up

Question 12

3.33 pts

Determine the concavity direction of the function at x = 4:

\(f(x) = -3x^{3}-x^{2}-3x+5\)

Concave down

Concave up

Inflection point

Cannot tell

Explanation:

Solution - Concavity at a Point

📌 The function:

\(f(x) = -3x^{3}-x^{2}-3x+5\)

📌 Step 1: First derivative

\(f'(x) = -9x^2-2x-3\)

📌 Step 2: Second derivative

\(f''(x) = -18x-2\)

📌 Step 3: Substitute x = 4 into the second derivative

\(f''(4) = -18 \cdot 4 + -2\)

\(f''(4) = -72 + -2\)

\(f''(4) = -74\)

📌 Step 4: Check the sign

\(f''(4) = -74\) < 0 (negative)

📌 Step 5: Conclusion

If f'(x) > 0	→	Concave up (∪)
If f'(x) < 0	→	Concave down (n)

Since \(f'(4) < 0 (negative)\), the function is concave concave down (∩)

✓ Answer: Concave down

Question 13

3.33 pts

Determine the concavity direction of the function at x = -2:

\(f(x) = -x^{3}+x^{2}+4x+3\)

Concave up

Concave down

Inflection point

Cannot tell

Explanation:

Solution - Concavity at a Point

📌 The function:

\(f(x) = -x^{3}+x^{2}+4x+3\)

📌 Step 1: First derivative

\(f'(x) = -3x^2+2x+4\)

📌 Step 2: Second derivative

\(f''(x) = -6x+2\)

📌 Step 3: Substitute x = -2 into the second derivative

\(f''(-2) = -6 \cdot -2 + 2\)

\(f''(-2) = 12 + 2\)

\(f''(-2) = 14\)

📌 Step 4: Check the sign

\(f''(-2) = 14\) > 0 (positive)

📌 Step 5: Conclusion

If f'(x) > 0	→	Concave up (∪)
If f'(x) < 0	→	Concave down (n)

Since \(f'(-2) > 0 (positive)\), the function is concave concave up (∪)

✓ Answer: Concave up

Question 14

3.33 pts

Find the inflection point of the function:

\(f(x) = -3x^{3}+5x^{2}-4x+4\)

\(x = \frac{5}{9}\)

\(x = \frac{-5}{9}\)

x = \(\frac{2}{5}\)

No inflection point

Explanation:

Solution - Inflection Point

📌 The function:

\(f(x) = -3x^{3}+5x^{2}-4x+4\)

📌 Step 1: First derivative

\(f'(x) = -9x^2+10x-4\)

📌 Step 2: Second derivative

\(f''(x) = -18x+10\)

📌 Step 3: Solve f'(x) = 0

\(-18x+10 = 0\)

\(x = \frac{5}{9}\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	\frac{5}{9}	...	+∞
f''(x)		+	0	-
Concavity		∪	Inflection	∩

📌 Step 5: Conclusion

There is a sign change in the second derivative around x = \frac{5}{9}

The concavity changes from ∪ to ∩

Therefore this is an inflection point.

✓ Answer: \(x = \frac{5}{9}\)

Question 15

3.33 pts

Find the inflection point of the function:

\(f(x) = -3x^{3}-6x^{2}-x\)

\(x = \frac{-2}{3}\)

\(x = \frac{2}{3}\)

x = \(\frac{-1}{12}\)

No inflection point

Explanation:

Solution - Inflection Point

📌 The function:

\(f(x) = -3x^{3}-6x^{2}-x\)

📌 Step 1: First derivative

\(f'(x) = -9x^2-12x-1\)

📌 Step 2: Second derivative

\(f''(x) = -18x-12\)

📌 Step 3: Solve f'(x) = 0

\(-18x-12 = 0\)

\(x = \frac{-2}{3}\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	\frac{-2}{3}	...	+∞
f''(x)		+	0	-
Concavity		∪	Inflection	∩

📌 Step 5: Conclusion

There is a sign change in the second derivative around x = \frac{-2}{3}

The concavity changes from ∪ to ∩

Therefore this is an inflection point.

✓ Answer: \(x = \frac{-2}{3}\)

Question 16

3.33 pts

Find the inflection point of the function:

\(f(x) = 2x^{3}-6x^{2}+3x-5\)

\(x = 1\)

\(x = -1\)

x = \(\frac{1}{4}\)

No inflection point

Explanation:

Solution - Inflection Point

📌 The function:

\(f(x) = 2x^{3}-6x^{2}+3x-5\)

📌 Step 1: First derivative

\(f'(x) = 6x^2-12x+3\)

📌 Step 2: Second derivative

\(f''(x) = 12x-12\)

📌 Step 3: Solve f'(x) = 0

\(12x-12 = 0\)

\(x = 1\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	1	...	+∞
f''(x)		-	0	+
Concavity		∩	Inflection	∪

📌 Step 5: Conclusion

There is a sign change in the second derivative around x = 1

The concavity changes from ∩ to ∪

Therefore this is an inflection point.

✓ Answer: \(x = 1\)

Question 17

3.33 pts

Find the inflection point of the function:

\(f(x) = x^{3}-6x^{2}+5x\)

\(x = 2\)

\(x = -2\)

x = \(\frac{5}{12}\)

No inflection point

Explanation:

Solution - Inflection Point

📌 The function:

\(f(x) = x^{3}-6x^{2}+5x\)

📌 Step 1: First derivative

\(f'(x) = 3x^2-12x+5\)

📌 Step 2: Second derivative

\(f''(x) = 6x-12\)

📌 Step 3: Solve f'(x) = 0

\(6x-12 = 0\)

\(x = 2\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	2	...	+∞
f''(x)		-	0	+
Concavity		∩	Inflection	∪

📌 Step 5: Conclusion

There is a sign change in the second derivative around x = 2

The concavity changes from ∩ to ∪

Therefore this is an inflection point.

✓ Answer: \(x = 2\)

Question 18

3.33 pts

Find the concavity intervals (concave up / concave down) of the function:

\(f(x) = -3x^{3}-3x^{2}-1\)

Up: \((-\infty, \frac{-1}{3})\), Down: \((\frac{-1}{3}, \infty)\)

Up: \((\frac{-1}{3}, \infty)\), Down: \((-\infty, \frac{-1}{3})\)

Concave up everywhere

Concave down everywhere

Explanation:

Solution - Concavity Intervals

📌 The function:

\(f(x) = -3x^{3}-3x^{2}-1\)

📌 Step 1: First derivative

\(f'(x) = -9x^2-6x\)

📌 Step 2: Second derivative

\(f''(x) = -18x-6\)

📌 Step 3: Find inflection points (f'(x) = 0)

\(-18x-6 = 0\)

\(x = \frac{-1}{3}\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	\frac{-1}{3}	...	+∞
f''(x)		+	0	-
Concavity		∪	Inflection	∩

📌 Step 5: Conclusion

• Concave up (∪) on the interval: \((-\infty, \frac{-1}{3})\)

• Concave down (n) on the interval: \((\frac{-1}{3}, \infty)\)

✓ Answer: Up: \((-\infty, \frac{-1}{3})\), Down: \((\frac{-1}{3}, \infty)\)

Question 19

3.33 pts

Find the concavity intervals (concave up / concave down) of the function:

\(f(x) = -3x^{3}+3x^{2}-2x+5\)

Up: \((-\infty, \frac{1}{3})\), Down: \((\frac{1}{3}, \infty)\)

Up: \((\frac{1}{3}, \infty)\), Down: \((-\infty, \frac{1}{3})\)

Concave up everywhere

Concave down everywhere

Explanation:

Solution - Concavity Intervals

📌 The function:

\(f(x) = -3x^{3}+3x^{2}-2x+5\)

📌 Step 1: First derivative

\(f'(x) = -9x^2+6x-2\)

📌 Step 2: Second derivative

\(f''(x) = -18x+6\)

📌 Step 3: Find inflection points (f'(x) = 0)

\(-18x+6 = 0\)

\(x = \frac{1}{3}\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	\frac{1}{3}	...	+∞
f''(x)		+	0	-
Concavity		∪	Inflection	∩

📌 Step 5: Conclusion

• Concave up (∪) on the interval: \((-\infty, \frac{1}{3})\)

• Concave down (n) on the interval: \((\frac{1}{3}, \infty)\)

✓ Answer: Up: \((-\infty, \frac{1}{3})\), Down: \((\frac{1}{3}, \infty)\)

Question 20

3.33 pts

Find the concavity intervals (concave up / concave down) of the function:

\(f(x) = -x^{3}-x^{2}\)

Up: \((-\infty, \frac{-1}{3})\), Down: \((\frac{-1}{3}, \infty)\)

Up: \((\frac{-1}{3}, \infty)\), Down: \((-\infty, \frac{-1}{3})\)

Concave up everywhere

Concave down everywhere

Explanation:

Solution - Concavity Intervals

📌 The function:

\(f(x) = -x^{3}-x^{2}\)

📌 Step 1: First derivative

\(f'(x) = -3x^2-2x\)

📌 Step 2: Second derivative

\(f''(x) = -6x-2\)

📌 Step 3: Find inflection points (f'(x) = 0)

\(-6x-2 = 0\)

\(x = \frac{-1}{3}\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	\frac{-1}{3}	...	+∞
f''(x)		+	0	-
Concavity		∪	Inflection	∩

📌 Step 5: Conclusion

• Concave up (∪) on the interval: \((-\infty, \frac{-1}{3})\)

• Concave down (n) on the interval: \((\frac{-1}{3}, \infty)\)

✓ Answer: Up: \((-\infty, \frac{-1}{3})\), Down: \((\frac{-1}{3}, \infty)\)

Question 21

3.33 pts

Determine the concavity direction of the function at x = -2:

\(f(x) = -2x^{3}+5x^{2}-4x-2\)

Concave up

Concave down

Inflection point

Cannot tell

Explanation:

Solution - Concavity at a Point

📌 The function:

\(f(x) = -2x^{3}+5x^{2}-4x-2\)

📌 Step 1: First derivative

\(f'(x) = -6x^2+10x-4\)

📌 Step 2: Second derivative

\(f''(x) = -12x+10\)

📌 Step 3: Substitute x = -2 into the second derivative

\(f''(-2) = -12 \cdot -2 + 10\)

\(f''(-2) = 24 + 10\)

\(f''(-2) = 34\)

📌 Step 4: Check the sign

\(f''(-2) = 34\) > 0 (positive)

📌 Step 5: Conclusion

If f'(x) > 0	→	Concave up (∪)
If f'(x) < 0	→	Concave down (n)

Since \(f'(-2) > 0 (positive)\), the function is concave concave up (∪)

✓ Answer: Concave up

Question 22

3.33 pts

Determine the concavity direction of the function at x = -2:

\(f(x) = x^{3}+2x^{2}-x-2\)

Concave down

Concave up

Inflection point

Cannot tell

Explanation:

Solution - Concavity at a Point

📌 The function:

\(f(x) = x^{3}+2x^{2}-x-2\)

📌 Step 1: First derivative

\(f'(x) = 3x^2+4x-1\)

📌 Step 2: Second derivative

\(f''(x) = 6x+4\)

📌 Step 3: Substitute x = -2 into the second derivative

\(f''(-2) = 6 \cdot -2 + 4\)

\(f''(-2) = -12 + 4\)

\(f''(-2) = -8\)

📌 Step 4: Check the sign

\(f''(-2) = -8\) < 0 (negative)

📌 Step 5: Conclusion

If f'(x) > 0	→	Concave up (∪)
If f'(x) < 0	→	Concave down (n)

Since \(f'(-2) < 0 (negative)\), the function is concave concave down (∩)

✓ Answer: Concave down

Question 23

3.33 pts

Find the inflection point of the function:

\(f(x) = 2x^{3}+2x^{2}+5x+1\)

\(x = \frac{-1}{3}\)

\(x = \frac{1}{3}\)

x = \(\frac{-5}{4}\)

No inflection point

Explanation:

Solution - Inflection Point

📌 The function:

\(f(x) = 2x^{3}+2x^{2}+5x+1\)

📌 Step 1: First derivative

\(f'(x) = 6x^2+4x+5\)

📌 Step 2: Second derivative

\(f''(x) = 12x+4\)

📌 Step 3: Solve f'(x) = 0

\(12x+4 = 0\)

\(x = \frac{-1}{3}\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	\frac{-1}{3}	...	+∞
f''(x)		-	0	+
Concavity		∩	Inflection	∪

📌 Step 5: Conclusion

There is a sign change in the second derivative around x = \frac{-1}{3}

The concavity changes from ∩ to ∪

Therefore this is an inflection point.

✓ Answer: \(x = \frac{-1}{3}\)

Question 24

3.33 pts

Find the inflection point of the function:

\(f(x) = -2x^{3}-x^{2}-2x+5\)

\(x = \frac{-1}{6}\)

\(x = \frac{1}{6}\)

x = -1

No inflection point

Explanation:

Solution - Inflection Point

📌 The function:

\(f(x) = -2x^{3}-x^{2}-2x+5\)

📌 Step 1: First derivative

\(f'(x) = -6x^2-2x-2\)

📌 Step 2: Second derivative

\(f''(x) = -12x-2\)

📌 Step 3: Solve f'(x) = 0

\(-12x-2 = 0\)

\(x = \frac{-1}{6}\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	\frac{-1}{6}	...	+∞
f''(x)		+	0	-
Concavity		∪	Inflection	∩

📌 Step 5: Conclusion

There is a sign change in the second derivative around x = \frac{-1}{6}

The concavity changes from ∪ to ∩

Therefore this is an inflection point.

✓ Answer: \(x = \frac{-1}{6}\)

Question 25

3.33 pts

Find the inflection point of the function:

\(f(x) = -x^{3}-5x+3\)

\(x = 0\)

\(x = 1\)

x = 5

No inflection point

Explanation:

Solution - Inflection Point

📌 The function:

\(f(x) = -x^{3}-5x+3\)

📌 Step 1: First derivative

\(f'(x) = -3x^2-5\)

📌 Step 2: Second derivative

\(f''(x) = -6x\)

📌 Step 3: Solve f'(x) = 0

\(-6x = 0\)

\(x = 0\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	0	...	+∞
f''(x)		+	0	-
Concavity		∪	Inflection	∩

📌 Step 5: Conclusion

There is a sign change in the second derivative around x = 0

The concavity changes from ∪ to ∩

Therefore this is an inflection point.

✓ Answer: \(x = 0\)

Question 26

3.33 pts

Find the inflection point of the function:

\(f(x) = -3x^{3}+4x^{2}-5\)

\(x = \frac{4}{9}\)

\(x = \frac{-4}{9}\)

x = 0

No inflection point

Explanation:

Solution - Inflection Point

📌 The function:

\(f(x) = -3x^{3}+4x^{2}-5\)

📌 Step 1: First derivative

\(f'(x) = -9x^2+8x\)

📌 Step 2: Second derivative

\(f''(x) = -18x+8\)

📌 Step 3: Solve f'(x) = 0

\(-18x+8 = 0\)

\(x = \frac{4}{9}\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	\frac{4}{9}	...	+∞
f''(x)		+	0	-
Concavity		∪	Inflection	∩

📌 Step 5: Conclusion

There is a sign change in the second derivative around x = \frac{4}{9}

The concavity changes from ∪ to ∩

Therefore this is an inflection point.

✓ Answer: \(x = \frac{4}{9}\)

Question 27

3.33 pts

Find the concavity intervals (concave up / concave down) of the function:

\(f(x) = -2x^{3}+6x^{2}-4x+4\)

Up: \((-\infty, 1)\), Down: \((1, \infty)\)

Up: \((1, \infty)\), Down: \((-\infty, 1)\)

Concave up everywhere

Concave down everywhere

Explanation:

Solution - Concavity Intervals

📌 The function:

\(f(x) = -2x^{3}+6x^{2}-4x+4\)

📌 Step 1: First derivative

\(f'(x) = -6x^2+12x-4\)

📌 Step 2: Second derivative

\(f''(x) = -12x+12\)

📌 Step 3: Find inflection points (f'(x) = 0)

\(-12x+12 = 0\)

\(x = 1\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	1	...	+∞
f''(x)		+	0	-
Concavity		∪	Inflection	∩

📌 Step 5: Conclusion

• Concave up (∪) on the interval: \((-\infty, 1)\)

• Concave down (n) on the interval: \((1, \infty)\)

✓ Answer: Up: \((-\infty, 1)\), Down: \((1, \infty)\)

Question 28

3.33 pts

Find the concavity intervals (concave up / concave down) of the function:

\(f(x) = -x^{3}-6x^{2}-2x-1\)

Up: \((-\infty, -2)\), Down: \((-2, \infty)\)

Up: \((-2, \infty)\), Down: \((-\infty, -2)\)

Concave up everywhere

Concave down everywhere

Explanation:

Solution - Concavity Intervals

📌 The function:

\(f(x) = -x^{3}-6x^{2}-2x-1\)

📌 Step 1: First derivative

\(f'(x) = -3x^2-12x-2\)

📌 Step 2: Second derivative

\(f''(x) = -6x-12\)

📌 Step 3: Find inflection points (f'(x) = 0)

\(-6x-12 = 0\)

\(x = -2\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	-2	...	+∞
f''(x)		+	0	-
Concavity		∪	Inflection	∩

📌 Step 5: Conclusion

• Concave up (∪) on the interval: \((-\infty, -2)\)

• Concave down (n) on the interval: \((-2, \infty)\)

✓ Answer: Up: \((-\infty, -2)\), Down: \((-2, \infty)\)

Question 29

3.33 pts

Find the concavity intervals (concave up / concave down) of the function:

\(f(x) = -3x^{3}-x-5\)

Up: \((-\infty, 0)\), Down: \((0, \infty)\)

Up: \((0, \infty)\), Down: \((-\infty, 0)\)

Concave up everywhere

Concave down everywhere

Explanation:

Solution - Concavity Intervals

📌 The function:

\(f(x) = -3x^{3}-x-5\)

📌 Step 1: First derivative

\(f'(x) = -9x^2-1\)

📌 Step 2: Second derivative

\(f''(x) = -18x\)

📌 Step 3: Find inflection points (f'(x) = 0)

\(-18x = 0\)

\(x = 0\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	0	...	+∞
f''(x)		+	0	-
Concavity		∪	Inflection	∩

📌 Step 5: Conclusion

• Concave up (∪) on the interval: \((-\infty, 0)\)

• Concave down (n) on the interval: \((0, \infty)\)

✓ Answer: Up: \((-\infty, 0)\), Down: \((0, \infty)\)

Question 30

3.33 pts

Find the concavity intervals (concave up / concave down) of the function:

\(f(x) = 3x^{3}+4x^{2}+3x-1\)

Up: \((\frac{-4}{9}, \infty)\), Down: \((-\infty, \frac{-4}{9})\)

Up: \((-\infty, \frac{-4}{9})\), Down: \((\frac{-4}{9}, \infty)\)

Concave up everywhere

Concave down everywhere

Explanation:

Solution - Concavity Intervals

📌 The function:

\(f(x) = 3x^{3}+4x^{2}+3x-1\)

📌 Step 1: First derivative

\(f'(x) = 9x^2+8x+3\)

📌 Step 2: Second derivative

\(f''(x) = 18x+8\)

📌 Step 3: Find inflection points (f'(x) = 0)

\(18x+8 = 0\)

\(x = \frac{-4}{9}\)

📌 Step 4: Sign table for f'(x)

x	−∞	...	\frac{-4}{9}	...	+∞
f''(x)		-	0	+
Concavity		∩	Inflection	∪

📌 Step 5: Conclusion

• Concave up (∪) on the interval: \((\frac{-4}{9}, \infty)\)

• Concave down (n) on the interval: \((-\infty, \frac{-4}{9})\)

✓ Answer: Up: \((\frac{-4}{9}, \infty)\), Down: \((-\infty, \frac{-4}{9})\)