Sketching Polynomials - Part 2 of 3

In this video, I demonstrate how to find the information we need to be able to sketch or roughly graph the function:

f(x) = 3x^4 + 4x^3 - 12x^2

The first step is to find the stationary or critical points.
These occur when the slope of the curve is equal to 0. To do this we, differentiate the original function to find the slope function, and set this equal to 0:

{f}'(x)=12x^3+12x^2-24x=0

The solutions to this equation will give us the x-coordinates of the critical points, which we can substitute into f(x) to find the y-coordinate.

The next step is to determine the nature of the critical points. To do this, we need to find the second derivative of f(x) by differentiating f'(x):

f''(x) = 36x^2 + 24x - 24

We then evaluate f''(x) at the x-coordinates of the critical points.

• If f'' is greater than 0, we have a local minimum
• If f'' is less than 0, we have a local maximum
• If f'' = 0, we have a point of inflection