Math Questions and Answers

Start Your Free Trial

Identify the open intervals on which the function f(x)=3x-x^3 is increasing or decreasing?

Expert Answers info

Tushar Chandra eNotes educator | Certified Educator

calendarEducator since 2010

write12,554 answers

starTop subjects are Math, Science, and Business

A function f(x) is increasing when f'(x) is positive and it is decreasing when f'(x) is negative.

f(x) = 3x - x^3

f'(x) = 3 - 3x^2

First determine where f'(x) = 0

3 - 3x^2 = 0

=> 1 - x^2 = 0

=> x^2 = 1

=> x = -1 and x = 1

We have the intervals (-inf., -1), (-1, 1) and (1, inf.)

In (-inf., -1), f'(x) is seen to be less than 0. Hence the unction is decreasing here. Similarly, the function is decreasing in (1, inf.). The function is increasing is (-1, 1).

The function is increasing in the interval (-1, 1) and decreasing elsewhere.

check Approved by eNotes Editorial



giorgiana1976 | Student

The sign of the 1st derivative tells us about the monotony of a function.

If the 1st derivative is strictly positive, then the function is strictly increasing and if the 1st derivative is strictly negative, then the function is strictly decreasing.

We'll differentiate the function with respect to x:

f'(x) = 3 - 3x^2

We'll detemrine the roots of f'(x):

3-3x^2 = 0

1 - x^2 =0

The difference of two squares will return the product:

(1-x)(1+x)=0

We'll cancel each factor:

1-x=0 =>x=1

1+x=0 => x=-1

The derivative is positive over (-1,1) and it is negative over the intervals (-`oo` ,-1) and (1,`oo` )

Therefore, the function is decreasing over the intervals (-`oo` ,-1) and (1,`oo` ) and it is increasing over the interval (-1,1).