Determine the concavity of $f(x)=x^4-10x^3$.

We fully admit that this may be hard, but it is worth understanding. Now what we mean by finding the concavity is finding the values of $x$ for which it is concave up and concave down.

The first thing we should do is graph the function to get an idea of its concavity. From the graph below it appears that the function is concave up for negative values of $x$. Then for some small values of positive $x$ it appears concave down and then for larger values of positive $x$ it is concave up again.

First, we know that $f\,''(x)=12x^2-60x$. The most helpful thing we can do is factor out the $x$ so $f\,''(x)=x(12x-60)$. We know to do this from experience. This is helpful because we want to know when $f\,''(x)$ is positive or negative so being able to see that $x$ is positive or negative easily will help because to determine if $f\,''(x)$ is positive or negative we will make use of the fact that a negative number time a positive number will be negative and two numbers that have the same sign and are multiplied together will always result in a positive number.

The first thing to note is that when $x\lt 0$ then $f\,''(x)\gt 0$. This is because $12x-60$ is negative and a negative number times a negative number ($x$) is always positive. So when $x$ is negative $f(x)$ is concave up.

Now recall that $12\cdot 5=60$. So when $x\gt 5$, then $12x-60\gt 0$ (positive) making $f\,''(x)\gt 0$. Therefore, when $x\gt 5$, $f(x)$ is concave up.

So all we have left is $0\lt x\lt 5$. In this situation $x$ is (obviously) positive and $12x-60$ is negative. This means that $f\,''(x)\lt 0$ and is concave down.