Let be a differentiable function on the closed interval with derivative whose graph…

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

Problem

Let

p

be a differentiable function on the closed interval

[-3,6]

with derivative

p'

whose graph is described below. • For

-3\le x\le1

, the graph of

p'

is the upper semicircle of radius

2

centered at

(-1,0)

. • For

1\le x\le6

, the graph of

p'

is the line segment joining the points

(1,2)

and

(6,-3)

. Assume

p(-3)=4

. (a) Evaluate

p(1)

and

p(6)

. Show your reasoning, including any geometric calculations used. (b) Determine the absolute minimum value of

p

[-3,6]

and the

x

–value at which it occurs. Justify fully, referring to critical points and endpoints as needed. (c) Discuss all intervals on

(-3,6)

where the concavity of

p

changes. Support your conclusions by analyzing how the slope of

p'

varies on each portion of its graph.