Video Lecture

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We have seen that derivatives represent the slope of the tangent line, i.e., the best approximation of the instantaneous rate of change of the function at a specific input. We used this to determine where functions are increasing or decreasing, and using this we can even find local extrema of functions.

The ability to find local maximums or minimums (a.k.a. extrema) is useful in many ways. From maximizing profits, to minimizing costs, to finding optimal solutions to complicated systems, it is arguably one of the most useful general skills covered in calculus.

So, how do we find these local extrema? First, let’s recall what exactly a local extrema is. Consider the following graph:

Local extrema are the highest or lowest values locally, in other words, you can choose an interval small enough around the point such that the point is either the largest or smallest y-value on the graph within that interval. In our example, we have two local extrema:

The dot on the left represents a local maximum (as we can see by looking the red segment ’nearby’ that spot, the entirety of which is lower than the dot) and the dot on the right represents a local minimum (as similarly shown by the blue segment). But, how do we find these values? The key lies in noticing something rather interesting about the tangent lines at these points:

Both tangent lines are horizontal! Indeed, the key idea here is to note that all local extrema have these horizontal tangent lines!

So, all we need to do is find where the tangent lines are horizontal, and that will gives us the extrema right? ... Unsurprisingly, it isn’t quite that simple - math never is right? Nonetheless, we do need to find where the tangent line is horizontal, that is the first step - it just isn’t going to be the only step.

To that end, notice that a horizontal line has a slope of zero. Thus in order to find the points where the tangent line is horizontal, we need to find where the slope of the tangent line is zero. If you don’t see why this works, you should view the above video/link for a nice animation to explain this better, but the short version is that if the slope wasn’t zero, then it would either be increasing or decreasing - but then we could move slightly to find a higher (or lower for a local minimum) point than the supposed local extrema, which isn’t possible!

So we believe that the local extrema occur where the derivative is zero, but as we’ve seen when we discussed continuity and differentiability, it’s important to ask if this is sufficient. In other words, if the derivative is zero at a point, does that necessarily mean that the point must be a local max or min? To answer this, consider the following graph of $f(x)=x^3$.

Since we know the function we can take the derivative (using our rule for polynomials). In particular $f'(x) = 3x^{3-1} = 3x^2$. This is clearly zero at $x=0$, but looking back at our graph and checking where $x=0$, we can see that it is obviously not a local extrema. Nonetheless we can see why the tangent line is horizontal - the graph appears to be flattening out briefly at the origin before continuing.

So, unfortunately, it is not enough to find the zeros of the derivative. Nonetheless, finding the points where the derivative equal zero is still the first step to finding local extrema - indeed, these points can be important for a number of reasons, and as such they get a special name.

But, as we saw, being a critical point isn’t enough - there are critical points that may not be local extrema. So how do we tell if a given critical point is a local extrema? Our answer is in the original animation we saw!

Calculus students are typically taught (and remember) that “the local extrema occur where the derivative is zero” - but really it’s more accurate to say that the “local extrema occur where the derivative transitions from one sign to the other” - either negative to positive or positive to negative. Thus, to find local extrema, we first find the critical points, then we use a sign chart to determine which of these points are actually transition points from negative to positive or positive to negative. If it is a transition from negative to positive, then the slope is transitioning from going down to going up - hence it is a minimum. Likewise if the slope is transitioning from positive to negative, then the slope is transitioning from going up to going down - so it is a maximum.

We have seen that local extrema occur where there is a horizontal tangent line, i.e. where the derivative is zero. We call the points with this feature (of having a horizontal tangent line) critical points, and we can find them by first finding where the derivative equals zero. But we also know that just finding the critical points isn’t enough - once we have those points, we must determine if they are actually local extrema. To do this, we can use a sign chart to determine if they are actually local extrema - and the sign chart also tells of which type of extrema they are at the same time.