A continuous function is graphed on the interval as described below
MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability
Problem
A continuous function g is graphed on the interval [−2,2] as described below.
• For −2≤x<0, the graph is a straight line segment of slope 2 that starts at (−2,−1) and ends at (0,3).
• At x=0, the point (0,3) is solid. For 0<x≤2, the graph follows the curve y=3+x2/3.
• As x approaches 2 from the left, the slope of the curve becomes unbounded, and the graph has a vertical tangent at (2,3+34).
Respond to the prompts below.
(a) Identify all x–values in [−2,2] at which g is NOT differentiable.
(b) For each x–value found in part (a), classify the specific geometric cause of non-differentiability (removable discontinuity, jump, corner, cusp, or vertical tangent).
(c) Justify each classification in part (b) by referencing the behavior of slopes of secant lines approaching the point from the left and from the right.