Local and Global Extrema

A local extrema is the lowest or highest point on one part of a graph.

For example, the quadratic $y=x^2$ has a local extrema at $(x,y)=(0,0)$ (because (0,0) is the lowest point on the graph).

Why do we care about minimum/maximum points?

If a function has a local minimum or maximum at $x=k$, then the derivative of that function evaluated at $x=k$ will be 0.

Take $y=x^2$. The derivative is $2x$, and when $x=0$, we have $2x=0$.

Therefore, we can find all the extreme points by taking the derivative of a function and setting it equal to 0.

The derivative of $f(x)$ is

$f'(x)=x^2+5x+6$, which has roots at $x=-2, -3$.

Plugging these in, we get $(x,y)=(-2,-\frac{2}{3})(-3,-\frac{1}{2})$ are the extreme points.

In fact, the first point turns out to be a local minima, and the second one a local maxima. You can see this on a graphing calculator.

(Video) What Does it Mean to be Increasing/Decreasing/A Local Max/A Local Min

by PatrickJMT

Here's a video by patrickJMT where he goes over what it means for a function to be increasing or decreasing and what it means for a point to be a local max or a local min.

Here's a dramatic reenactment of the graph that Patrick uses in his video.

Notice that when we move left to right along our graph, the $y$ values get smaller and smaller until they reach point $A$. What we would say is that on the interval of negative infinity to negative three the function is decreasing. $$(-\infty,-3)\ \searrow$$

Notice that from point $A$, $x=-3$, to point $B$, $x=1$, the $y$ values are getting bigger and bigger. We would say the function is increasing on this interval. $$(-3,1)\ \nearrow$$

From $x=1$ to $x=5$ the function is a flat line. So we say the function is constant on this interval. But then after that it starts decreasing again. $$(5,6)\ \searrow$$

Then it increases for a bit. $$(6,7)\ \nearrow$$

Finally, it decreases one last time and continuous decreasing. $$(7,\infty)\ \searrow$$

So this is all that's meant when you're asked where a function increases or decreases.

Local Minimums or Relative Minimums are essentially the bottoms of the valleys on the graph. The technical definition says the height of a local minimum has to be less than or equal to the heights of the points around it.

Similarly, Local Maximums or Relative Maximums are the tops of the hills on the graph. The technical definition says the height of a local maximum has to be greater than or equal to the heights of the points around it.

So let's look at point $A$ on the graph. It's at the bottom of a valley. If we look at $A$'s height we see it is less than or equal to the heights of the points around it. Therefore we can conclude that point $A$ is a local minimum.

If we look at point $F$ we see that the height of the points to the left of it are smaller and the height of the points to the right of it are larger. Therefore, it is neither a local minimum or a local maximum.

Now look at $B$. It's sitting at the top of a hill. The height of the points to the left of it are smaller and the height of the points to the right of it are the equal. Because it's height has to be greater than or equal to to be a local maximum we can conclude that $B$ is a local max.

$C$ is a special point. Because the points around it are the same height it fits into the definition of a local maximum. But by that same logic it fits into the definition of a local minimum. So point $C$ is a local max AND a local min.

Hopefully you can tell that point $D$ is a local minimum and point $E$ is a local maximum.