Decide if the functions are differentiable at x=0 . Try zooming in on a graphing calculator, or

step 1 For the given problem, we want to decide if the functions are differentiable at x equals 0. So w

Decide if the functions are differentiable at x=0 . Try...

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Key Concepts

Absolute Value Function

The absolute value function introduces potential non-smoothness because it may create a sharp corner or cusp in a graph, notably at the point where the argument of the absolute value is zero. Understanding the behavior of |x| is key to analyzing the differentiability of functions that incorporate it.

Differentiability

Differentiability of a function at a point implies that the function has a well-defined, unique tangent line there, which occurs when both the left-hand and right-hand derivatives exist and are equal. This concept is essential in determining the smoothness of a function and its behavior at specific points.

Piecewise Analysis

When a function involves expressions like |x|, it is often necessary to consider its behavior separately over different intervals—typically where the argument is non-negative versus negative—to compute one-sided derivatives. This approach helps in evaluating the overall differentiability of the function at points where the definition changes.

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