Question

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

Answer 1

concave at x = 1

Explanation:

To determine if f(x) is concave/convex at x = a , we consider the value of f''(a).

• If f''(a) > 0 , then f(x) is convex at x = a

• If f'')a) < 0 , then f(x) is concave at x = a

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

differentiate using the `color(blue)"power rule"`

`rArrf'(x)=-3x^2+4x-2`

and `f''(x)=-6x+4`

`rArrf''(1)=-6+4=-2`

Since f''(1) < 0 , then f(x) is concave at x = 1
graph{-x^3+2x^2-2x-2 [-10, 10, -5, 5]}