Define the function p(x)= x(6-x) + 1 x-2

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

Problem

Define the function

p(x)=\sqrt{x(6-x)}+\frac{1}{x-2},

with domain consisting of all real numbers

x

such that

0\le x\le 6

and

x\ne 2

. (a) Justify the stated domain of

p

. (b) Compute

p'(x)

and solve

p'(x)=0

for all values in the domain. (c) Identify all values in the domain where

p'(x)

is undefined while

p

is defined. (d) Create a sign chart for

p'(x)

on its domain, indicating each interval of increase or decrease. (e) Determine every relative (local) maximum and minimum value of

p

on the closed interval

[0,6]

, and justify each conclusion. Include an analysis of the endpoints

x=0

and

x=6