Let f(x)=|x^3-9 x|. a. Does f^'(0) exist? b. Does f^'(3) exist? c. Does f^'(-3) exist? d. Determ

step 1 Suppose f of x equals the absolute value of x cubed minus 9x. Then if you take the derivative of

Let f(x)=|x^3-9 x|. a. Does f^'(0) exist? b. Does f^'(3)...

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Key Concepts

Critical Points and Extrema in Calculus

Finding extrema of functions involves locating critical points where the derivative is zero or does not exist, in addition to verifying endpoints or boundary behavior. This process is central in calculus for optimization and understanding the overall behavior of functions.

Differentiability at Non-smooth Points

Differentiability of a function requires that the left-hand and right-hand derivatives at a point are equal. For functions involving absolute values, potential points of non-differentiability occur where the inner function is zero, since the absolute value can introduce corners or cusps where the derivative might not exist.

Piecewise Function Representation

Absolute value functions can be rewritten in a piecewise form based on the sign of the inner expression. This representation allows one to analyze different regions separately, making it easier to study properties such as continuity and differentiability, as well as to compute derivatives.

Absolute Value Function

An absolute value function is one that takes the non?negative value of its argument. Its behavior is characterized by a change of slope at points where the expression inside the absolute value is zero, which can lead to issues with differentiability if the left-hand and right-hand derivatives do not match.

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