Question

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

Answer 1

`f(x)` is concave DOWNWARDS (convex?) at `x=1`.

Explanation:

This problem may be easier if we determine the expanded (not factored) form of `f(x)` first.

`f(x)=x(x-2)(x+3)(x-1)=x(x^2+x-6)(x-1)=x(x^3-7x+6)=x^4-7x^2+6x`

`f^'(x)=4x^3-14x+6`

and

`f^('')(x)=12x^2-14`

`f^('')(1)=12*1^2-14=-2`

Which means that `f(x)` is concave DOWNWARDS (convex?) at `x=1` because `f^('')(1)<0`

graph{x(x-2)(x+3)(x-1) [-4, 3, -26, 10]}