Question

Is `f(x)=(x-3)(x-2)-x^2+x^3` concave or convex at `x=-1`?

Answer 1

Concave down

Explanation:

To determine if a function is concave up or down at a point, you need to examine the sign of the second derivative `f''(x)` at that point.

First, multiply out and simplify `f(x)`:

`f(x) = x^2-2x-3x+6-x^2+x^3`
`f(x) = x^3 -5x + 6`

Find `f'(x)` and then `f''(x)`:

`f'(x) = 3x^2 - 5`
`f''(x) = 6x`

At `x=-1`, `f''(-1) = -6 < 0`. Since the second derivative is negative at `x=-1`, that indicates that the method is concave down.

graph{(x-3)(x-2)-x^2+x^3 [-14.32, 14.15, -3.2, 11.04]}