Mathematics

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

. a. (i) Find all values of

x

at which

h'(x)=0

h'(x)

is not defined. b. (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. c. (iii) Examine whether

x=-3,\;x=0

and

x=3

can be points of local extremum of

h

, giving full justification for each case.

Intuitive explanation

The key is recognizing that you can recover $h$ from its derivative by integration, and the continuity condition at $x=0$ pins down the constant of integration. Since $h'(x)$ is defined separately on $(-\infty,0)$ and $(0,\infty)$ , each piece will have its own integration constant—but continuity at the boundary forces them to be equal. Integrate $h'(x)$ on each interval, then use $h(0)=4$ to find the unique constant that makes the function continuous everywhere.

Intuitive explanation

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