Question

For what values of x is `f(x)=(x-1)(x-7)(x-1)` concave or convex?

Answer 1

Convex if x is greater than 1.5, concave if less than 1.5

Explanation:

A function is convex (aka concave up) when `f''(x)>0`, and concave aka concave down when <0. Thus, we must find the second derivative. We first multiply these factors...

`f (x)=(x-1)(x-7)(x-1) = (x^2-2x+1)(x-7) = x^3 - 7x^2 - 2x^2 +14x +x-7 = x^3 - 9x^2 + 15x -7 = f (x)`

Now we differentiate using the power rule...

`f'(x) =3x^2-9x+15`

And once more...

`f''(x) = 6x -9`

We then find when this is greater than 0...

`6x-9>0 -> 6x>9 -> x>1.5`

This, the function is convex for x greater than 1.5, concave for x less than 1.5