A function is defined by f(x)= cases 2-x, & x<2, (x-2)^2+k, & x 2

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

A function

f

is defined by

f(x)=\begin{cases} 2-x, & x<2,\\ (x-2)^2+k, & x\ge 2, \end{cases}

where

k

is a real constant. Consider the interval

[-1,5]

. (a) Determine the value of

k

that makes

f

continuous at

x=2

. (b) With this value of

k

, compute

f(-1)

and

f(5)

. (c) Justify that the equation

f(x)=4

has at least one real solution on

[-1,5]

. State an open sub-interval of

[-1,5]

that must contain such a solution.