Question

the graph of f is given. select the x - values at which f is not differentiable with the correct reason(s). (description: graph comprised of two components. first component rises in the 2nd quadrant, turns sharply downward at (-2, 4) continues to fall intersecting y = 2 to an open dot at (1, 1). the second component begins with a solid dot at (1, -2.5) and rises. it rises vertically through x = 3 and continues to rise through the 1st quadrant. ) 3, function not defined -2, function not defined 1, function not defined 3, vertical tangent 1, discontinuous

Explanation:

Step1: Recall differentiability conditions

A function is not differentiable at points of discontinuity, sharp - corners, or where there is a vertical tangent.

Step2: Analyze \(x = - 2\)

At \(x=-2\), the function has a sharp corner. The slope of the function changes abruptly from positive (as it was increasing in the second - quadrant) to negative (as it starts decreasing after the corner). So, the function is not differentiable at \(x = - 2\) due to the corner.

Step3: Analyze \(x = 1\)

The function has a jump - discontinuity at \(x = 1\). The left - hand limit and the right - hand limit of the function as \(x\to1\) are not equal. The left - hand limit approaches \(y = 1\) (open dot) and the right - hand limit approaches \(y=-2.5\) (solid dot). Since the function is discontinuous at \(x = 1\), it is not differentiable at \(x = 1\).

Step4: Analyze \(x = 3\)

The function has a vertical tangent at \(x = 3\). As \(x\) approaches \(3\), the slope of the secant lines approaches infinity. So, the function is not differentiable at \(x = 3\) because of the vertical tangent.

Answer:

-2, corner
1, discontinuous
3, vertical tangent