AP

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

Problem

Let

f(x) = \frac{1}{3}x^3 + x^2 - 15x + 17

. (1) Find

f'(x)

. (2) The graph of

f

has horizontal tangents at the points where

x = a

and

x = b

a < b

. Find the value of

a

and the value of

b

. (3) Sketch the graph of

y = f'(x)

. (4) Hence explain why the graph of

f

has a local maximum point at

x = a

. (5) Find

f''(b)

. (6) Hence, use your answer to part (d)(i) to show that the graph of

f

has a local minimum point at

x = b

. (7) The normal to the graph of

f

x = a

and the tangent to the graph of

f

x = b

intersect at the point

(p, q)

. Find the value of

p

and the value of

q