Let the function be defined on by f(x)= cases x^ 2 , & x 1, 2x-1, & x>1

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

Problem

Let the function

f

be defined on

\mathbb{R}

f(x)=\begin{cases} x^{2}, & x\le 1,\\ 2x-1, & x>1. \end{cases}

Answer the following questions about

f

. (a) Show that

f

is continuous at

x=1

. (b) Compute the one–sided derivatives of

f

x=1

, and state whether

f

is differentiable there. (c) Decide whether

f

has a relative (local) maximum, a relative (local) minimum, or neither at

x=1

. Provide a calculus–based justification that refers to the behavior of

f'(x)

on intervals immediately to the left and right of

x=1

. Explain your reasoning clearly.