39-42 The graph of $f$ is given. State, with reasons, the numbers at which $f$ is not differentiable.
To state: The graph of the function f is not differentiable at which the numbers.
The function f is not differentiable at −4 and 0.
Note: The function f is not differentiable at the point a, then it must satisfies any of the following conditions.
(i) The function f is discontinuous at the point a.
(ii) The function f has a corner point at the point a.
(iii) The function f has a vertical tangent at the point a.
From the graph of f, it is observed that f has a corner point at x=−4. That is, f has no tangent at that point and it is not differentiable at x=−4.
The graph of f has a jump discontinuity at x=−4. That is, the limit does not exist because the left and right hand limits are not equal as x approaches 0.
Therefore, f is discontinuous at x=−4.
Thus, it can be concluded that, f is not differentiable at the point x=0.