Consider a differentiable function on with derivative h'(x)= x^ 2 -9 x

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

Problem

Consider a differentiable function

h

(-\infty,0)\cup(0,\infty)

with derivative

h'(x)=\frac{x^{2}-9}{x},\qquad x\ne0.

Assume

h

is continuous at

x=0

and

h(0)=4

. (i) Find all values of

x

at which

h'(x)=0

h'(x)

is not defined. (ii) Determine the sign of

h'(x)

on each of the intervals

(-\infty,-3),\;(-3,0),\;(0,3)

and

(3,\infty)

and hence state where

h

is increasing and where it is decreasing. (iii) Examine whether

x=-3,\;x=0

and

x=3

can be points of local extremum of

h

, giving full justification for each case.