Determine the maxima and minima of the function $f(x)=x^4-10x^3$.

The first step is to graph the function. It appears that at $x=0$ there is neither a maximum nor minimum and at $x=7.5$ there is a minimum.

The second step to finding the maxima and minima is to find all of the critical points. We have done this already for this function here. So our critical points are at $x=0$ and $x=7.5$. We need to investigate each of them individually.

Before we get started though we will also need the second derivative if we want to use the second derivative test. This is $f\,''(x)=12x^2-60x$.

Let's look at $x=0$ first. Let's look on either side of $x=0$. For example let's try looking at $x=-1$ and $x=1$. $f\,'(1)=-34$ and $f\,'(1)=-26$. The slope around $x=0$ does not go from negative to positive so it is not a minimum and it does not go from positive to negative so it is not a maximum. Also, $f\,''(0)=0$. In other words, it is not greater than or less than zero. So the critical point at $x=0$ is neither a maximum nor a minimum.

Next, let's look at $x=7.5$. Let's try looking at $x=7.25$ and $x=7.75$. At $x=7.25$ $f\,'(7.25)=-52.5625$ and at $x=7.75$ $f\,'(7.75)=60.0625$. So the derivative goes from negative to positive. This means there is a minimum at $x=7.5$. Let's also check the second derivative. At $x=7.5$ $f\,''(7.5)=225$ which means that it is a minimum.

Both our first and second derivative tests were consistent for both $x=0$ and $x=7.5$. There is neither a maximum nor minimum at $x=0$ and there is a minimum at $x=7.5$.