Let the function be defined on the interval by f(x)= cases (x+2)^2,& -2 x 2, 6x-x^ 2…

MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Let the function

f

be defined on the interval

-2\le x\le 6

f(x)=\begin{cases}(x+2)^2,& -2\le x\le 2,\\6x-x^{2},& 2<x\le6.\end{cases}

Define the accumulation function

g(x)=\int_{0}^{x} f(t)\,dt,

for all

x

[-2,6]

. (a) Compute

f'(x)

on each sub-interval of the domain. (b) Determine the sign of

f'(x)

on the intervals

(-2,2)

and

(2,6)

. Provide supporting work. (c) Using your conclusions in part (b), indicate the intervals on which

g

is concave up and the intervals on which

g

is concave down. (d) Identify all

x

-coordinates in the open interval

(-2,6)

at which the graph of

g

could have a point of inflection. For each such

x

-value, justify whether a point of inflection actually occurs.