Critical Points Calculator

Q: What is a critical point?

A point in the domain where f'(x) = 0 or f'(x) does not exist. Every local maximum or minimum of a differentiable function happens at a critical point, but not every critical point is an extremum.

Q: How do I tell a maximum from a minimum?

First derivative test: if f' changes from + to − it is a local max; from − to + a local min; no sign change means neither. Second derivative test: f''(c) 0 means min, and f''(c) = 0 is inconclusive.

Q: What about absolute (global) extrema on a closed interval?

Evaluate f at every critical point inside [a, b] and at both endpoints. The largest value is the absolute maximum and the smallest is the absolute minimum (the closed interval method). The table below does exactly this.

Find every x where the slope is zero, then classify each as a local maximum, minimum or neither, with a sign chart for f′ and the absolute extrema on your interval.

Function f(x)

Reads as

x^{3} - 6 x^{2} + 9 x + 1

Try:

Interval fromto

Critical points

$x = 1$ local maximum
$x = 3$ local minimum

f
f′

f′ > 0 ↗f′ < 0 ↘f′ > 0 ↗

Step by step

Differentiate
$f'(x) = 3 x^{2} - 12 x + 9$
Solve f′(x) = 0
Searching on [-1, 5].
$x = 1,\quad x = 3$
Classify x = 1
First derivative test: local maximum. Second derivative test: f″ = -6 < 0 → max.
$f(1) = 5$
Classify x = 3
First derivative test: local minimum. Second derivative test: f″ = 6 > 0 → min.
$f(3) = 1$

Closed interval method
$x$	type	$f(x)$
-1	endpoint · absolute min	-15
1	critical	5
3	critical	1
5	endpoint · absolute max	21

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Finding and classifying critical points

Differentiate: find $f'(x)$ (see the derivative calculator).
Solve $f'(x) = 0$ , and note where $f'$ is undefined but $f$ is defined.
Make a sign chart for $f'$ . Where the sign flips from + to −, the graph goes up then down: a local maximum. From − to +: a local minimum.
Or use the second derivative: $f''(c) > 0$ means the curve is cupped upward there (a minimum).

A critical point with no sign change, like $x = 0$ for $x^3$ , is neither. It often turns out to be an inflection point instead.

Questions students ask

What is a critical point?

A point in the domain where $f'(x) = 0$ or $f'(x)$ does not exist. Every local maximum or minimum of a differentiable function happens at a critical point, but not every critical point is an extremum.

How do I tell a maximum from a minimum?

First derivative test: if $f'$ changes from + to − it is a local max; from − to + a local min; no sign change means neither. Second derivative test: $f''(c) < 0$ means max, $f''(c) > 0$ means min, and $f''(c) = 0$ is inconclusive.

What about absolute (global) extrema on a closed interval?

Evaluate $f$ at every critical point inside $[a, b]$ and at both endpoints. The largest value is the absolute maximum and the smallest is the absolute minimum (the closed interval method). The table below does exactly this.

Why does it search only in a window?

Critical points are found numerically on the interval you choose and then snapped to exact values where possible. Functions like $\sin x$ have infinitely many, so a finite window is needed.