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