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Critical Point of a Function

Tool to find the critical points of a function, corresponding to the critical values where the derivative is zero or not defined.

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# Critical Point of a Function

## Critical Point Calculator

Tool to find the critical points of a function, corresponding to the critical values where the derivative is zero or not defined.

### What is a critical point? (Definition)

A critical point is a point of a function where the gradient is zero or not defined (the derivative is equal to 0 or the derivative is not real). A critical point is similar to a stationary point (except for the undefined part) its value maybe maximum / minimum local / global.

### How to calculate critical points?

From the function $f$, calculate its derivative $f '$ and look at the critical values for which it cancels $f'(x) =$ 0 or the values for which it is not defined (see domain derivability).

Example: The square root function $f(x) = \sqrt{x}$ has for derivative $f'(x) = \frac{1}{2\sqrt{x}}$ which is not defined (over the reals) for $x <= 0$, its critical values are therefore all negative numbers (including 0).

### What is the difference between a critical point and a stationary point?

A critical point is the union of all the points where the derivative is zero (called stationary points) with all the points or the derivative is not defined (called singular points).

So all stationary points are critical points but not all critical points are stationary points.

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