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

Sep 17, 2017

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

#### Explanation:

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

$f \left(x\right) = \left(x - 1\right) \left(x - 7\right) \left(x - 1\right) = \left({x}^{2} - 2 x + 1\right) \left(x - 7\right) = {x}^{3} - 7 {x}^{2} - 2 {x}^{2} + 14 x + x - 7 = {x}^{3} - 9 {x}^{2} + 15 x - 7 = f \left(x\right)$

Now we differentiate using the power rule...

$f ' \left(x\right) = 3 {x}^{2} - 9 x + 15$

And once more...

$f ' ' \left(x\right) = 6 x - 9$

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

$6 x - 9 > 0 \to 6 x > 9 \to x > 1.5$

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