Lesson 5. Max / Min and Slope Zero

Reminder

Our primary goal at this moment in learning calculus is to find maximum / minimum of certain functions!

Slope at max / min

Last time, we have considered the continuous function $f(x) = -x^{2} + 1$. Using FooPlot, we have observed that the function takes maximum at $x = 0$ with the value $f(0) = 1$. Here, we make an important observation. Again, type "-x^2 + 1" under "Function $y(x)$" on FooPlot. It gives you all the points $(x, y)$ such that $y = -x^{2} + 1$. Pick any point on the curve $y = -x^{2} + 1$. At each point, you can draw a line that meets that curve only at that point but nowhere else. Do you follow? Otherwise, please ask questions using the comment section below!

Such a line is called the tangent line at the point you picked. Now, look at the slope of the tangent line as you move the point along the curve $y = -x^{2} + 1$. What do you see? When does the slope get steep and when does it become flat? You can notice that at $x = 0$, where the function $f(x) = -x^{2} + 1$ has maximum, the slope is completely flat! We call this "slope $0$".

What is going on?

How did you get slope $0$ at the point where the function has maximum? Is this a mystery? No! Notice that maximum means highest point when you think of the graph of the function as a mountain. So when you go to the mountain, you go up before the maximum and go down after the maximum.

Be careful: your maximum may have a sharp cusp!

On FooPlot, type "1 - sqrt(x^2)" under "Function $y(x)$", it gives you the graph $y = 1 - |x|$. Don't be scared. The symbol $|x|$ means the distance between $x$ and $0$. Let's try some numbers, we have

$|0| = 0$,
$|1| = 1$,
$|\pi| = \pi$.

But we also have

$|-1000| = 1000$,
$|-\pi^{10}| = \pi^{10}$.

Do you have some feeling about $|x|$ now? Otherwise, don't forget to ask questions using the comment section below!

Now, go back to the plot $y = 1 - |x|$ you drew. Where is the maximum? The maximum of the function $x \mapsto 1 - |x|$ happens at $x = 0$, and the value of it is $y = 1$. However, what is the slope at $x = 0$? This is impossible to tell! If you look at the left side of $x = 0$, you would say you have slope $1$. However, if you look at the right side of $x = 0$, you would say you have slope $-1$. (Remember, from math classes at school back in the days, slope means rise/run!)

This is somewhat frustrating. For the function $f(x) = 1 - x^2$, we observed that we had slope $0$ at maximum because we go up and then turn back down at the highest point. However, for the function $g(x) = 1 - |x|$, this was not true! We could not even talk about slope at $x = 0$ because there were two different slopes based on whether we come from the left or the right side of $x = 0$.

The reason for this issue is that for $g(x) = 1 - |x|$, when we reach the top of the mountain (or the maximum at $x = 0$), some steep jump in our direction happens. That is, the graph $y = 1 - |x|$ is NOT smooth at $x = 0$. It turns out that as long as your graph is smooth, this issue does not happen.

Moral. When a function $f(x)$ reaches a maximum or a minimum, as long as the function is smooth, it must have slope $0$ at that point.