Let f(x) = x^3 − 8x^2. Find the points on the graph of f where the tangent line is horizontal.

Question

I found the derivative and the smaller x-value which is (0,0), I just can't figure out how to solve for the larger x-value!

Answer 1

Well, if you got the derivative as 3x² - 16x, you should be home free.

When you factor the derivative you get x(3x-16)=0

The 2 roots are x=0 (which you already got) and x=16/3.

Answer 2

Let f(x)=x^3-8x^2. Calculate the derivative of f(x), f'(x)=3x^2-16x=x(3x-16). Solve f'(x)=0 to get x=0 or x=3/16 and those are the only two points where the tangent line at the graph is horizontal.