# Derivative - determining intervals of increase and decrease

• Mar 29th 2008, 03:12 PM
theowne
Derivative - determining intervals of increase and decrease
Given a function h = g(f(x)) where f = (x+1)^2 and g = 1/x, determine the intervals in which the function increases.

I tried to find the derivative function and got -2(x+1)^-3 which seems right. What I"m not sure is how to approach determining the intervals. My initial thought is to determine where the derivative touches zero and then substitute values on either side to determine whether the value, and therefore the slope of the original function, is positive or negative. However that would be like this:

0 = -2(x+1)^-3

And I'm not sure how I could solve that for x in that form. Can someone provide some nudging in the right direction?
• Mar 29th 2008, 03:21 PM
mr fantastic
Quote:

Originally Posted by theowne
Given a function h = g(f(x)) where f = (x+1)^2 and g = 1/x, determine the intervals in which the function increases.

I tried to find the derivative function and got -2(x+1)^-3 which seems right. What I"m not sure is how to approach determining the intervals. My initial thought is to determine where the derivative touches zero and then substitute values on either side to determine whether the value, and therefore the slope of the original function, is positive or negative. However that would be like this:

0 = -2(x+1)^-3

And I'm not sure how I could solve that for x in that form. Can someone provide some nudging in the right direction?

A function h(x) is increasing when $\displaystyle h'(x) > 0$ and decreasing when $\displaystyle h'(x) < 0$ ....
• Mar 29th 2008, 03:23 PM
mr fantastic
Quote:

Originally Posted by mr fantastic
A function h(x) is increasing when $\displaystyle h'(x) > 0$ and decreasing when $\displaystyle h'(x) < 0$ ....

Alternatively, for this simple case $\displaystyle h(x) = \frac{1}{(x+1)^2}$ you could just draw the graph .... Clearly it's increasing for x < -1 and decreasing for x > -1.
• Mar 29th 2008, 03:27 PM
theowne
Yes, but is there a way to tell by using the derivative function, exactly for what intervals of x, h'(x) is greater than 0 or less than 0/describe on paper.
• Mar 29th 2008, 03:34 PM
mr fantastic
Quote:

Originally Posted by theowne
Yes, but is there a way to tell by using the derivative function, exactly for what intervals of x, h'(x) is greater than 0 or less than 0/describe on paper.

You solve the inequalities I stated in post #2, which in this case are:

Increasing: $\displaystyle \frac{-2}{(x+1)^3} > 0 \Rightarrow (x + 1)^3 < 0$, and

Decreasing: $\displaystyle \frac{-2}{(x+1)^3} < 0 \Rightarrow (x + 1)^3 > 0$.