For the function, find all critical points or determine that no such points exist.
f (x) = 4x3 + 3x2 – 2x + 1
We find the derivative:
f '(x) = 12x2 + 6x – 2.
Since this is a polynomial, it's defined everywhere. In order to find the roots of the derivative, first factor out the unnecessary 2:
f '(x) = 2(6x2 + 3x – 1).
Then we need to use the quadratic formula:
These are the critical points, and our exact answers. However, to check our work on the graph it's useful to know approximate values:
If we graph the function f using marks every 0.25 on the x-axis, we do appear to have horizontal tangents around 0.25 and -0.75. This is enough evidence to support our answer.