Finding Concavity

Question

I'm trying to find the concavity and inflection points to this equation:

$f(x) = 3x^3 - x^2 + 2x - 7$

I've taken the derivative which is: $9x^2 - 2x + 2$, but it's not factorable so how would I find the concavity of this question?

Answer 1

To study the concavity, you need the second derivative of $f(x)$, which is easy to find. It is a linear function, and you will find it straightforward to determine where it changes from negative to positive.

Remark: If you wanted to determine local maxima and minima (which you were not asked about), then indeed you would be interested in the solutions of $f'(x)=0$. You could then use the Quadratic Formula to find the roots of $f'(x)$. In our particular case, $f'(x)=0$ has no real solutions.

One cannot expect that all quadratics will factor nicely: most of them don't. Soon, if not already, it will be taken for granted that you know the Quadratic Formula.

Answer 2

There is a formula to find the roots of a second order polinomial. Use it to factorize the derivative.