Let be the piecewise–defined function f(x)= cases x^ 2 +4x+5, & 0 x 3, 20-2x, & 3< x 8

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let

f

be the piecewise–defined function

f(x)=\begin{cases} x^{2}+4x+5, & 0\le x\le 3,\\ 20-2x, & 3< x\le 8. \end{cases}

The domain of

f

is the closed interval

[0,8]

. (a) For each piece, compute

f'(x)

. Identify every value of

x

(0,8)

at which

f'(x)=0

. (b) Create a complete list of all candidates for absolute extreme values of

f

[0,8]

. Briefly explain why no other points need to be considered. (c) Evaluate

f

at every candidate found in part (b) and at both endpoints of the domain. (d) State the absolute maximum and absolute minimum values of

f

[0,8]

, including the

x

–coordinates at which they occur.