The graph of a function is described on as follows
MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability
Problem
The graph of a function h is described on −4≤x≤4 as follows.
• On −4≤x<−2 the graph is a smooth decreasing curve.
• On −2≤x<0 the graph is a straight line of slope 0.
• At x=0 the point (0,1) is solid, and the left and right pieces meet there, forming a sharp corner.
• On 0<x<3 the graph is a smooth increasing curve whose slope approaches +∞ as x→3−.
• At x=3 the point (3,h(3)) is solid, and for 3<x≤4 the graph is a smooth decreasing curve.
Use this information to analyze h.
(a) State the set of all x–values in [−4,4] for which h is differentiable.
(b) On each open interval where h is differentiable, determine whether h is increasing or decreasing. Support your answer with reasoning based on slopes of tangent lines.
(c) Explain why h cannot possess a relative maximum at x=3, even though the derivative fails to exist there.