Question

For what values of x is `f(x)=-x^3-4x^2+2x+5` concave or convex?

Answer 1

The curve is convex for `x > -4/3` and concave for `x<-4/3`

Explanation:

Use the second differential to decide concavity. If `f''(x) > 0` then the curve is convex, and if `f''(x) < 0` then it is concave. Convex means the gradient is decreasing (going from positive to negative), and concave means the gradient is increasing (going form negative to positive).
`f'(x) = -3x^2 -8x +2`
`f''(x) = -6x -8`
`-6x - 8 > 0 iff x > 8/-6 iff -4/3`
The curve is convex for `x > -4/3` and concave for `x<-4/3`