For the function, find all critical points or determine that no such points exist.

*f* (*x*) = 4*x*^{3} + 3*x*^{2} – 2*x* + 1

Answer

*f* (*x*) = 4*x*^{3} + 3*x*^{2} – 2*x* + 1

We find the derivative:

*f *'(*x*) = 12*x*^{2} + 6*x* – 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(6*x*^{2} + 3*x* – 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.