A twice-differentiable function is defined on with the following properties

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

Problem

A twice-differentiable function

f

is defined on

0\le x\le7

with the following properties. •

f(0)=0

. • For

0<x<2

f

is increasing and its slope is increasing. • On

2<x<5

f

continues to increase, but its slope is decreasing and becomes zero at

x=5

. • For

5<x<7

f

is decreasing and its slope is becoming more negative. Let

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

for

0\le x\le7

. (a) On the interval

0<x<7

, indicate where the slopes of

f

are positive, zero, or negative. Justify each classification using the descriptive information above. (b) Determine the intervals on which

g

is concave up and the intervals on which

g

is concave down. Provide clear reasoning that connects your answer in part (a) to the concavity of

g

. (c) Identify all

x

-values in

(0,7)

at which the graph of

g

must have a point of inflection, and justify your conclusion. (d) Sketch a possible graph of

g

0\le x\le7

that is consistent with all of your conclusions. Clearly label the points of inflection and indicate concavity on your sketch.