Question

How do you find all points of inflection of the function `f(x)=x^3-3x^2-x+7`?

Answer 1

The inflection of a function means the changes in its direction. You **can either understand it as the change from increasing to decreasing (and vice-versa) on your `f(x)` or the vertex of the slope for your `f'(x)`.

But how do we find inflection points? Using your function's second derivatives `f''(x)`.

Just a quick reminder: your `f'(x)` measures the increase / decrease rate of `f(x)`, while `f''(x)` indicates whether `f(x)` changes from convex to concave (or vice-versa).

Thus, we must equal the second derivative of the function to zero.

`f'(x)=3x^2-6x-1`,
`f''(x)=6x-6`.

Equaling it to zero, we have `6x=6`, `x=1`.

For `f(1)=1^3-3(1^2)-1+7=4`

Thus, your inflection point is `(1,4)`

graph{x^3-3x^2-x+7 [-15.27, 16.33, -2.72, 13.08]}