Topic 5.2 Notes – Extreme Value Theorem, Global Versus Local Extrema, and Critical Points

1. The Extreme Value Theorem

The Extreme Value Theorem (EVT) is a guarantee statement.

If a function $f$ is continuous on a closed interval $[a,b]$, then:

• $f$ has at least one absolute maximum on $[a,b]$
• $f$ has at least one absolute minimum on $[a,b]$

It guarantees existence. It does not tell you where the extrema are.

What must be true

Two conditions are required:

• The interval is closed → written with brackets $[a,b]$, so endpoints are included.
• The function is continuous on the entire interval.

If either fails, EVT does not apply.

Here’s what that looks like visually. Notice how the example on the closed interval $[1,4]$ has both an absolute maximum and minimum, while the open interval example $[0,2)$ does not have an absolute maximum.

Examples of when absolute extrema do and do not exist

What EVT does NOT say

• It does not say there is only one max or min.
• It does not tell you whether they occur at endpoints or inside.
• It does not apply to open intervals like $(a,b)$.
• It does not apply if there’s a discontinuity (hole, jump, asymptote).

How to justify EVT on an FRQ

When you’re asked to justify existence, you must clearly say:

1. $f$ is continuous on $[a,b]$.
2. $[a,b]$ is closed and bounded.
3. Therefore, by the Extreme Value Theorem, $f$ has an absolute maximum and minimum on $[a,b]$.

If continuity is not stated or cannot be verified, you cannot claim the guarantee.

2. Global vs Local Extrema

Now let’s separate the types of extrema.

Absolute (Global) Maximum

• The highest value of the function on the entire interval.
• Can occur at a critical point or an endpoint.
• Guaranteed to exist if EVT conditions are met.

Absolute (Global) Minimum

• The lowest value on the entire interval.
• Also can occur at a critical point or an endpoint.

Local (Relative) Maximum

• Higher than nearby points.
• Only compares to values in a small neighborhood.
• Not required to be the highest overall.

Local (Relative) Minimum

• Lower than nearby points.
• Only compares locally.

Here’s the clean comparison:

Important detail students miss: endpoints can be absolute extrema, but they are not local extrema because there aren’t points on both sides.

3. Critical Points

A critical point occurs at $x=c$ (in the domain of $f$) where:

• $f'(c)=0$, or
• $f'(c)$ does not exist.

These are the only possible locations of local extrema.

Critical points on a graph of $f$

Why they matter

• Every local max or min occurs at a critical point.
• Not every critical point is an extremum.

In the graph above, notice the local minimum at $b$ and the local maximum at $d$, where $f'(x)=0$. Also notice the point at $c$. The tangent is horizontal there, but the function keeps increasing. That point is critical, but it is not an extremum.

A horizontal inflection point is the classic example. The derivative is zero, but the function does not change direction.

Finding critical points

If you’re given $f(x)$:

1. Compute $f'(x)$.
2. Solve $f'(x)=0$.
3. Find where $f'(x)$ is undefined.
4. Keep only values in the domain of $f$.

If you’re given a graph of $f'(x)$, look for where it equals zero or is undefined.

4. Finding Absolute Extrema on a Closed Interval

When the problem actually asks you to find them, use the Closed Interval Method every time.

1. Find critical points in $(a,b)$.
2. Evaluate $f$ at each critical point.
3. Evaluate $f(a)$ and $f(b)$.
4. Compare all values.

Largest value → absolute max.
Smallest value → absolute min.

Skipping endpoints is one of the most common point-loss mistakes on tests.

5. Common Mistakes and Exam Traps

• Applying EVT on an open interval.
• Claiming existence without mentioning continuity.
• Ignoring endpoints when asked for absolute extrema.
• Assuming every critical point is a max or min.
• Confusing local and absolute language in written answers.