Question

How do you find the inflections points for `f(x) = (1/3)x^3-x^2-8x+2`?

Answer 1

The unique inflection point is `(x,y)=(1,-20/3)`.

Explanation:

The first derivative is `f'(x)=x^2-2x-8` and the second derivative is `f''(x)=2x-2`.

The second derivative changes sign (from negative to positive) at `x=1`, so this is the first coordinate of the unique inflection point of `f` (where, in this case, the graph of `y=f(x)` changes from concave down to concave up).

The second coordinate of the unique inflection point is `f(1)=1/3-1-8+2=-20/3`.

Therefore, the unique inflection point is `(x,y)=(1,-20/3)`.