Question

How do you find the x and y coordinates of all inflection points `f(x) = x^4 - 12x^2`?

Answer 1

The coordinates of the two inflection points are `(x,y)=(pm sqrt(2),-20)`

Explanation:

The first derivative is `f'(x)=4x^3-24x` and the second derivative is `f''(x)=12x^2-24=12(x^2-2)`.

The second derivative is zero only at `x=pm sqrt(2)` and, in fact, changes sign as `x` increases through these two values. Therefore the `x`-coordinates of the two inflection points are `x=pm sqrt(2)`.

Since `f(pm sqrt(2))=4-12*2=4-24=-20`, it follows that the coordinates of the two inflection points are `(x,y)=(pm sqrt(2),-20)`.

Here's the graph. See if you can find the inflection points in the graph:

graph{x^4-12x^2 [-10, 10, -40,40]}