Solved Math Exams

Lesson

What Are Minimum and Maximum Values?

A function has a local maximum at a point if that point is higher than all the nearby points — it's a peak.

A function has a local minimum at a point if that point is lower than all the nearby points — it's a valley.

Together, these are called local extrema (singular: extremum).

Key Definitions

Local Maximum at $x = c$:

$f$ has a local maximum at $c$ if $f(c) \geq f(x)$ for all $x$ in some open interval around $c$.

Local Minimum at $x = c$:

$f$ has a local minimum at $c$ if $f(c) \leq f(x)$ for all $x$ in some open interval around $c$.

Key point: We only compare to nearby points, not the entire domain.

Visualizing Local Extrema

In this graph, notice the function $f(x) = x^3 - 3x$ has:

A local maximum at $x = -1$ (the peak)
A local minimum at $x = 1$ (the valley)

At both of these points, the tangent line is horizontal — the slope is zero.

The Connection to Derivatives

At a local maximum or minimum (in the interior of the domain), something special happens:

The tangent line is horizontal — meaning $f'(c) = 0$

This makes sense geometrically:

Just before a peak, the function is increasing ($f' > 0$)
Just after a peak, the function is decreasing ($f' < 0$)
At the exact peak, the slope must be zero ($f' = 0$)

Critical Points: Where to Look

A critical point occurs where:

$f'(c) = 0$ (horizontal tangent), OR
$f'(c)$ does not exist (corner, cusp, or vertical tangent)

Important: Not every critical point is a local extremum!

Example 1: Why $f'(c) = 0$ Doesn't Guarantee an Extremum

Consider $f(x) = x^3$. Find where $f'(x) = 0$ and determine if there's a local extremum.

Step 1: Find critical points

$$f'(x) = 3x^2$$ $$3x^2 = 0$$ $$x = 0$$

So $f'(0) = 0$. Is $x = 0$ a local max or min?

Step 2: Analyze the behavior

For $x < 0$: $f(x) = x^3 < 0$
For $x > 0$: $f(x) = x^3 > 0$
At $x = 0$: $f(0) = 0$

The function goes from negative to positive, passing through zero. It doesn't have a peak or valley — it just flattens out momentarily!

Step 3: Conclusion

$$\boxed{x = 0 \text{ is NOT a local extremum}}$$

Even though $f'(0) = 0$, the point $(0, 0)$ is neither a local max nor a local min. It's an inflection point where the function levels off but keeps going in the same direction.

Notice how $f(x) = x^3$ has a horizontal tangent at $x = 0$, but it's not a peak or valley — the function is always increasing!

Example 2: A Corner Creates an Extremum

Consider $f(x) = |x|$. Does $f$ have a local minimum at $x = 0$?

At $x = 0$, $f(0) = 0$.

For any $x \neq 0$, $f(x) = |x| > 0$.

So $f(0) \leq f(x)$ for all $x$ near $0$.

$$\boxed{\text{Yes, } f \text{ has a local minimum at } x = 0}$$

Note: Here $f'(0)$ does not exist (the graph has a sharp corner), but it's still a local minimum! Critical points include places where the derivative doesn't exist.

Example 3: Identifying Extrema from Derivative Sign

Given that:

$f'(x) > 0$ for $x < -1$
$f'(x) < 0$ for $-1 < x < 2$
$f'(x) > 0$ for $x > 2$

Identify all local extrema.

At $x = -1$: $f'$ changes from positive to negative

Function goes from increasing to decreasing
This is a local maximum

At $x = 2$: $f'$ changes from negative to positive

Function goes from decreasing to increasing
This is a local minimum

$$\boxed{\text{Local max at } x = -1, \text{ local min at } x = 2}$$

How to Classify Critical Points

$f'$ changes from...	Type of extremum
$+$ to $-$	Local maximum
$-$ to $+$	Local minimum
$+$ to $+$ or $-$ to $-$	Neither (not an extremum)

Common Mistakes and Misunderstandings

❌ Mistake: Assuming every critical point is an extremum

Wrong: "$f'(c) = 0$, so there must be a local max or min at $c$."

Why it's wrong: The derivative can be zero at an inflection point where the function just flattens out momentarily.

Example: $f(x) = x^3$ has $f'(0) = 0$, but $x = 0$ is NOT an extremum.

Correct: Check if $f'$ actually changes sign at the critical point.

❌ Mistake: Forgetting about corners and cusps

Wrong: "I set $f'(x) = 0$ and found all the extrema."

Why it's wrong: Local extrema can also occur where $f'$ doesn't exist — at corners or cusps.

Example: $f(x) = |x|$ has a local min at $x = 0$, but $f'(0)$ doesn't exist.

Correct: Check both where $f'(x) = 0$ AND where $f'(x)$ doesn't exist.

Formulas & Reference

Local Maximum Definition

f(c) \geq f(x) \text{ for all } x \text{ near } c

A function has a local maximum at x = c if the function value at c is greater than or equal to function values at all nearby points. The point is a 'peak' in the graph.

Variables:

$f(c)$:: The function value at the potential maximum
$f(x)$:: Function values at nearby points
$c$:: The point where the local maximum occurs

Local Minimum Definition

f(c) \leq f(x) \text{ for all } x \text{ near } c

A function has a local minimum at x = c if the function value at c is less than or equal to function values at all nearby points. The point is a 'valley' in the graph.