Question

Consider the function f(x)=x3+4x2+2. find the points where the tangent line is horizontal.

Answer 1

Thinking about the graph of f(x), the rate of change is zero whenever there is an extremum.

First, differentiate f(x)

$f'(x)=3x^2+8x$

Next, find the zeros of f'(x) by factoring.

$x(3x+8)=0 \\ x= \frac{-8}{3},x=0$

Now, we substitute these x-values into the original equation to find the coordinates. 

(-2.667, 11.481), (0, 2)