Question

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

Answer 1

for `x< -17/68` convexity and `x > -17/68` concavity

Explanation:

For a twice continuous function like the one proposed, the concavity or convexity is determined by the second derivative sign.
`d^2/(dx^2)f(x)=72x+68`, If `d^2/(dx^2)f(x) < 0` the curvature is considered as convex because the area region contained below is a convex set. If `d^2/(dx^2)f(x) > 0` is concave. Solving for `d^2/(dx^2)f(x) = 0` we get `x =-17/68` so for `x<-17/68` we have convexity and for `x>-17/68` concavity.