Question

question 1 the graph of a function \\( f(x) \\) is shown below: for how many values of \\( x \\) is \\( f(x) \\) not differentiable on the interval \\( (-2, 5) \\)? \\( \bigcirc 2 \\) \\( \bigcirc 5 \\) \\( \bigcirc 3 \\) \\( \bigcirc 1 \\) \\( \bigcirc 4 \\)

Explanation:

Step1: Recall Differentiability Rules

A function is not differentiable at a point if: 1) It is discontinuous (jump, removable, infinite discontinuity), 2) It has a corner/cusp, 3) It has a vertical tangent.

Step2: Identify Critical Points on Graph

• Point \( x = -1 \): Removable discontinuity (open circle), so not differentiable.
• Point \( x = 1 \): Infinite discontinuity (vertical asymptote), so not differentiable.
• Point \( x = 2 \): Jump discontinuity (open circle vs closed dot), so not differentiable.
• Point \( x = 4 \): Corner (change in slope abruptly), so not differentiable.

Step3: Count Non - Differentiable Points

Count the points \( x = -1, x = 1, x = 2, x = 4 \). There are 4 such points.

Answer:

4