Question

select all the points at which the graph above is not differentiable
□ -4
□ -3
■ -2
□ -1
□ 0
□ 1
■ 2
□ 3
□ 4

Explanation:

Step1: Recall non - differentiability conditions

A function is not differentiable at a point if there is a sharp corner, a cusp, a vertical tangent, or a discontinuity at that point.

Step2: Analyze the graph at \(x=-2\)

At \(x = - 2\), there is a sharp corner. The slope of the function changes abruptly at this point, so the left - hand derivative and the right - hand derivative are not equal. Thus, the function is not differentiable at \(x=-2\).

Step3: Analyze the graph at \(x = 2\)

At \(x = 2\), there is a discontinuity (a hole in the graph). A function must be continuous at a point to be differentiable there. Since it is not continuous at \(x = 2\), it is not differentiable at \(x = 2\).

Step4: Analyze other points

At \(x=-4,-3,-1,0,1,3,4\), the graph is smooth and has no sharp corners, cusps, vertical tangents, or discontinuities. So the function is differentiable at these points.

Answer:

-2, 2