How To Determine Whether A Function Is Increasing Or Decreasing Using Its Derivative – Math Guide

critical points

f'(x) is either zero or undefined

Critical points are points on a graph where the derivative, or slope of the curve, is either zero or undefined. They are important because they provide information about the behavior of the function in that interval.

There are different types of critical points:
– A local maximum occurs at a critical point where the derivative changes from positive to negative.
– A local minimum occurs at a critical point where the derivative changes from negative to positive.
– A point of inflection occurs at a critical point where the second derivative changes sign.

To find critical points, you need to take the derivative of the function and set it equal to zero. Solving for the variable will give you the x-value(s) of the critical point(s). You should also check the value of the second derivative at each critical point to determine the type of critical point it is.

It is important to note that not all critical points will lead to local maximums, local minimums, or points of inflection. A critical point could also be a point where the function changes direction without reaching a maximum or minimum. Additionally, a critical point that occurs at the endpoint of the domain is not considered a local maximum or local minimum.

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