Question

the graph of f is given. state the numbers at which f is not differentiable. (enter your answers as a comma-separated list.)

Explanation:

Step1: Recall non - differentiability conditions

A function \(y = f(x)\) is not differentiable at a point if:

1. The function is not continuous at that point.
2. The function has a corner (or cusp) at that point.
3. The function has a vertical tangent at that point.

Step2: Analyze the graph for \(x=-4\)

At \(x = - 4\), we need to check the behavior of the graph. Looking at the graph, there is a corner (a sharp turn) at \(x=-4\). A function with a corner at a point does not have a well - defined derivative at that point because the left - hand and right - hand derivatives are not equal.

Step3: Analyze the graph for \(x = 0\)

At \(x=0\), there is a discontinuity (a hole and a point). Since the function is not continuous at \(x = 0\) (the limit as \(x\) approaches \(0\) does not equal the function value at \(x = 0\)), the function is not differentiable at \(x=0\).

Step4: Analyze the graph for \(x = 2\)

At \(x = 2\), there is a sharp turn (a corner) or a point where the graph has a non - smooth transition. Also, we can check the continuity and the derivative existence. The function has a change in the slope abruptly at \(x = 2\), so the derivative does not exist at \(x=2\).

Answer:

\(-4,0,2\)