Problem

Let

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

. (a) Find

f'(x)

. The graph of

f

has horizontal tangents at the points where

x=a

and

x=b

a<b

. (b) Find the value of

a

and the value of

b

. (c.i) Sketch the graph of

y=f'(x)

. (c.ii) Hence explain why the graph of

f

has a local maximum point at

x=a

. (d.i) Find

f''(b)

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

f

has a local minimum point at

x=b

. (e) 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