James F. answered • 06/20/14

Statistics Graduate Student and Tutor

When we're interested in increasing/decreasing, we have to take a look at the first derivative.

f(x) = 4x^3 + 5 ---> f'(x) = 12x^2

By setting f'(x) = 0 and solving for x, we can search for a relative min/max.

f'(x) = 12x^2 = 0 ---> x = 0, so x is either a relative min or max

Next, pick a point less than 0 and greater than 0 to see the value of f'(x). If it switches from negative to positive, it's a minimum. If it switches from positive to negative, it's a maximum. If it does neither, then it's not a maximum or minimum.

Also, by checking values on either side of x = 0, you will determine if it is increasing or decreasing. For example, f'(2) = 12*2^2 = 48 > 0, so it is increasing.

I hope this can get you started!