Pre-Analysis — Extrema from a Graph

Pre-Analysis — Extrema from a Graph. Practice questions to deepen understanding of extrema from a graph in pre-analysis — without differentiation. Online math practice with full solutions and step-by-step explanations.

Pre-analysis extrema practice — local and global maxima and minima, identification from a graph, understanding without derivatives. Local/global maximum/minimum.

Explanation:

🔝 Local maximum

Definition:

Local maximum point:

A point \((x_0, f(x_0))\) such that \(f(x_0)\) is the highest value "in a neighborhood"

Not necessarily the highest in the entire graph!

What does it look like?

Like a "peak" 🏔️ on the graph

The function increases up to the point ↗
reaches a peak 🔝
and then decreases ↘

Example:

On the inverted parabola \(f(x) = -x^2\)

At \(x=0\): local maximum

\(f(0) = 0\)

That is the highest point!

Why "local"?

Because it is the highest in its area
not necessarily on the entire graph

Explanation:

🔻 Local minimum

Definition:

Local minimum point:

A point \((x_0, f(x_0))\) such that \(f(x_0)\) is the lowest value "in a neighborhood"

Not necessarily the lowest in the entire graph!

What does it look like?

Like a "valley" 🏞️ on the graph

The function decreases down to the point ↘
reaches a low 🔻
and then increases ↗

Example:

On the parabola \(f(x) = x^2\)

At \(x=0\): local minimum

\(f(0) = 0\)

That is the lowest point!

Why "local"?

Because it is the lowest in its area
not necessarily on the entire graph

Explanation:

🌍 Global maximum

Global maximum:

The highest point on the entire graph!

There is no higher point anywhere

Also called: absolute maximum

Local maximum:

The highest point in an area

There can be higher points elsewhere

Example:

Mount Everest 🏔️:
• Global maximum in the world
• The highest mountain!

Mount Meron 🗻:
• Local maximum in Israel
• The highest in its area
• but not the highest in the world

Note:

A global maximum is also a local maximum!

but a local maximum is not necessarily global

Explanation:

📊 Identifying a maximum

The rule:

A local maximum point:

Before the point: increasing ↗
At the point: peak 🔝
After the point: decreasing ↘

How to check?

1️⃣ Look to the left of the point
→ Is the graph increasing? ↗

2️⃣ Look to the right of the point
→ Is the graph decreasing? ↘

3️⃣ Both true?
→ This is a local maximum point! 🔝

Example:

Graph:
increasing on \((-\infty, 3)\)
decreasing on \((3, \infty)\)

At \(x=3\): local maximum ✓

Explanation:

📊 Identifying a minimum

The rule:

A local minimum point:

Before the point: decreasing ↘
At the point: low 🔻
After the point: increasing ↗

How to check?

1️⃣ Look to the left of the point
→ Is the graph decreasing? ↘

2️⃣ Look to the right of the point
→ Is the graph increasing? ↗

3️⃣ Both true?
→ This is a local minimum point! 🔻

Example:

Graph:
decreasing on \((-\infty, -2)\)
increasing on \((-2, \infty)\)

At \(x=-2\): local minimum ✓

Explanation:

🔢 Number of extrema

The answer:

There can be as many as you like!

• zero extrema
• one point
• two points
• ten points
• infinitely many points!

Examples:

Zero points:

\(f(x) = x\) (a straight line)

only increases, no extremum

One point:

\(f(x) = x^2\) (parabola)

one minimum at \(x=0\)

Two points:

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

one maximum and one minimum

Infinitely many points:

\(f(x) = \sin(x)\)

maximum and minimum every \(2\pi\)

Explanation:

📍 Point vs value

Maximum point:

An ordered pair: \((x_0, f(x_0))\)

includes both the \(x\) and the \(y\)

Example: \((3, 5)\)

Maximum value:

Only the number: \(f(x_0)\)

Only the \(y\) (the height)

Example: \(5\)

Full example:

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

There is a maximum at \(x=3\)
\(f(3) = -9 + 18 - 5 = 4\)

Maximum point: \((3, 4)\)
Maximum value: \(4\)

In everyday language:

Maximum point = "where and what"
Maximum value = "how much"

Explanation:

🎯 Endpoint

The answer: yes!

An endpoint of the domain can be a global extremum

even though the monotonicity does not change there

Example:

\(f(x) = x\) on the domain \([0, 5]\)

The function is only increasing ↗

but:
• At \(x=0\): global minimum (\(f(0)=0\))
• At \(x=5\): global maximum (\(f(5)=5\))

Why?

Because these are the extreme values in the domain!

Lowest: \(0\)
Highest: \(5\)

Note:

These are not local extremum points
(because the monotonicity does not change)

but they are global extrema in the domain!

Explanation:

❌ A common mistake!

Do not confuse extremum with sign!

The problem:

What the student thought:

"\(f(2)=10\) is positive
positive = high
so this is a maximum"

❌ That is incorrect!

✓ The correct view:

Maximum = highest in the area

It does not matter whether positive or negative!

You need to check the monotonicity:
• Increasing before?
• Decreasing after?

Counterexample:

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

There is a maximum at \(x=0\)
\(f(0) = -5\)

The value is negative! ✓
but still a maximum! ✓

because this is the highest point on the graph

Remember:

Extremum = relative location
Sign = location relative to the \(x\)-axis

Explanation:

📚 Summary - extremum points

🔝 Local maximum:

• Highest in the area
• Increase → decrease
• "Peak" on the graph 🏔️

🔻 Local minimum:

• Lowest in the area
• Decrease → increase
• "Valley" on the graph 🏞️

🌍 Global extremum:

• Highest/lowest on the entire graph
• Can also be at an endpoint of the domain

How to find?

1️⃣ Check the monotonicity
2️⃣ Look for changes:
• ↗→↘ = maximum
• ↘→↗ = minimum
3️⃣ Also check the endpoints of the domain

⚠️ Remember:

Extremum ≠ sign!

• A maximum can be negative
• A minimum can be positive

Important: relative location in the area!