Question

How do you find the horizontal tangent line of y(x)=x^3+6x^2+6 ?

Answer 1

Horizontal tangent lines occur when the derivative is zero.

If \(y(x)=x^3+6x^2+6\), then \(y'(x)=3x^2+12x\). For what values of \(x\) is \(y'=0\)?

Answer 2

Solve y'(x)=0 for x. If there are real roots, each such root represents the location (x value only) of a horizontal tangent line. If you find the corresponding y value, you will have found the point of tangency.