Define the piecewise function q(x)= cases x^ 2 -4 x-2 , & x 2

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

Problem

Define the piecewise function

q(x)=\begin{cases} \dfrac{x^{2}-4}{x-2}, & x\ne 2,\\ k, & x=2, \end{cases}

where

k

is a constant to be chosen. Consider the closed interval

[0,4]

. (1) Part 1: (a) Determine the value of

k

that makes

q

continuous at

x=2

. Show your work. (2) Part 2: (b) Verify that, with your chosen value of

k

, all hypotheses of the Extreme Value Theorem are satisfied for

q

[0,4]

. (3) Part 3: (c) Find the absolute maximum and absolute minimum values of

q

[0,4]

and the corresponding

x

-values where they occur. Justify your answer by analyzing critical points and endpoints.