Mathematics

Problem

Let

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

. a. Find

f'(x)

. b. 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

. c. Sketch the graph of

y = f'(x)

. d. Hence explain why the graph of

f

has a local maximum point at

x = a

. e. Find

f''(b)

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

f

has a local minimum point at

x = b

. g. 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