Critical Points

Where is a function largest? Where is it smallest? To answer these questions you don’t need to check every point. Peaks and valleys can only happen where the graph levels off or has a sharp point, and those places are called critical points. Finding them is the first step of optimization, curve sketching and the first and second derivative tests that come next in this chapter.

What Is a Critical Point?

A critical point of is a number in the domain of where

There are two kinds:

• Stationary points, where . The tangent line is horizontal there.
• Singular points, where doesn’t exist but does: corners, cusps and vertical tangent lines.

The words “in the domain” matter. A value where itself is undefined, like for , is never a critical point.

Why Critical Points Matter

A local maximum is a point that is higher than every point near it, and a local minimum is lower than every point near it. At a smooth peak or valley, the tangent line has to be flat: if the slope were positive, the graph would still be rising, and if it were negative, it would be falling. That’s Fermat’s theorem:

If has a local maximum or minimum at , and exists, then .

If doesn’t exist, the extreme value can be at a corner or cusp, like the bottom of . Put together: every local maximum and minimum happens at a critical point. So to find them, you only need to check the critical points, not every .

The reverse is not true. A critical point is only a candidate. In the picture above, has , but the graph flattens out for a moment and keeps rising: no maximum, no minimum. Deciding which candidates are real extremes is the job of the first and second derivative tests.

How to Find Critical Points

1. Find the domain of .
2. Find and simplify it, ideally into a factored fraction.
3. Solve . For a fraction, that means numerator .
4. Find where is undefined. For a fraction, that means denominator .
5. Keep only values in the domain of . Those are the critical points.

Example 1: A Cubic

Determine all critical points of .

Step 1 — differentiate and factor:

Step 2 — solve :

Step 3 — check for undefined points. is a polynomial, so it’s defined everywhere. Polynomials only ever have stationary critical points.

The critical points are and , at the points and . The graph shows a peak at the first and a valley at the second.

Example 2: A Critical Point That Isn’t an Extreme

Find the critical points of .

Both are critical points, but they behave differently. Near , the factor is positive on both sides and is negative, so on both sides: the function falls, flattens for an instant, and falls again. At , changes from negative to positive, so the graph turns around at a minimum, .

A squared factor like in is a warning sign: touches zero there without changing sign.

Example 3: Where the Derivative Is Undefined

Find the critical points of .

The numerator is never 0, so there are no stationary points. The denominator is 0 at , where is undefined, and is defined. So is a critical point: the cusp at the bottom of the graph, which is a minimum.

Example 4: Both Kinds at Once

Find the critical points of .

Step 1 — expand so the power rule applies:

Step 2 — differentiate:

Step 3 — factor out the smaller power, , to get one fraction:

Step 4 — read off both kinds. The numerator is 0 at . The denominator is 0 at , and is defined. So the critical points are (a vertical tangent) and (the minimum, ).

Factoring into a single fraction is what makes both kinds visible. Leaving as two separate terms makes it easy to miss .

Example 5: A Rational Function

Find the critical points of .

By the quotient rule:

The numerator is 0 at . The denominator is never 0. The critical points are and , where and .

Example 6: Check the Domain

Find the critical points of .

is never 0, and it’s undefined at . But isn’t in the domain of , so it can’t be a critical point. has no critical points. The same applies to vertical asymptotes in general: the derivative is undefined there, but so is the function.

Example 7: Trig and Exponential Functions

Find the critical points of on .

Dividing by is safe here: where , , so .

Find the critical points of . The product rule gives

Since is never 0, the only critical point is .

What Comes Next: Classifying Critical Points

Finding critical points gives you the candidates. The next lessons decide what each one is:

• The first derivative test checks whether changes sign at : from to is a maximum, from to a minimum, no change means neither. That’s the same sign reasoning as increasing and decreasing intervals, now done with the derivative.
• The second derivative test uses the sign of , as in the higher-order derivatives lesson.
• On a closed interval, the extreme value theorem adds the endpoints to the list of candidates.

Common Mistakes

• Only solving . Points where is undefined count too. In Example 4, skipping them misses .
• Including points outside the domain. is not a critical point of or .
• Assuming every critical point is a maximum or minimum. at 0 is a critical point and neither.
• Solving instead of . That finds x-intercepts, not critical points.
• Dividing away a factor that can be zero. Factor instead: has the solution , which disappears if you divide by .

Practice Problems

Try each one before opening the answer. The practice set has 12 more with instant checking.

Problem 1. Find the critical points of .

Problem 2. Find the critical points of .

Problem 3. Find the critical points of .

Problem 4. Find the critical points of .

Problem 5. Find the critical points of .

Problem 6. Find the critical points of on .

Critical points tell you where to look. Next, the extreme value theorem uses them to find the absolute highest and lowest values of a function on a closed interval.