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

Answers to Questions

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).

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