A piecewise–linear graph of is described as follows: • For

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

A piecewise–linear graph of

u=g(x)

is described as follows: • For

0\le x\le2

, the graph is a straight line from

(0,0)

(2,3)

. • For

2\le x\le5

, the graph is a straight line from

(2,3)

(5,0)

. Let

y=f(u)

where

f(u)=u^{3}-4u

. Consider the composite function

y=f(g(x))

. (a) Compute

\displaystyle \lim_{x\to2^-}\dfrac{dy}{dx}

. (b) Compute

\displaystyle \lim_{x\to2^+}\dfrac{dy}{dx}

. (c) Determine whether

y

is differentiable at

x=2

. Provide a clear justification based on your answers to parts (a) and (b). (d) Is

y

continuous at

x=2

? Verify your conclusion with appropriate reasoning.