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Difference between local and absolute minimum and maximum (Let's make it easy for Donald Trump)

Alright Donald. Let's get started.

Let's say Jacob deGrom threw a rubber ball (not a baseball this time) and after hitting the ground and bouncing in the air, the height (y) of that ball at any given time (x) takes the shape of the graph below:

A and B are the two points of maxima (plural of maximum) and C and D are the two points of minima (plural of minimum) in this graph. In this hills and valley graph the maximums and minimums are found by looking at where the curve flattens. The only exception here is D which is called the saddle point. Let's start with the maxima (the two maximum points).

Of A and B which one is the local maximum and which one is the absolute maximum?

B is the local maximum while A is the absolute maximum. Why?

Let's understand LOCAL MAXIMUM first. Read slowly Donald.

LOCAL MAXIMUM AND ABSOLUTE MAXIMUM

B is the local maximum because it is at the highest height of a function (the graph above is a function) given a particular domain that does not cover the entire graph. This is called an open interval where the end points of the entire graph are not included. Ok wait. What is domain? Don't worry Donald Trump. Lets make it simpler.

Look at the graph below:

The domain or in other words the 'input values on the x axis' of the entire graph is roughly between 0 and 10 (where the green curve starts and ends). Between 0 and 10 on the x axis you have two maxima, A and B. But if you just choose a domain; say between 6 and 9 (the two red lines) then B is the only maximum. Now that is called the local maximum or relative maximum. This is the maximum only between the domain we were given. This means only between the given times 6 and 9 the ball reached a maximum height at B.

Mathematically, it could be written in this general format:

The graph is f(x) and the domain is x1<x<x2. Hence f(a) where 'a' is the value on the x axis at the local maximum B (a=7.5) is greater than or equal to all other maximums within this domain.

You can identify 'a' in the graph given below (where the yellow line touches the x axis).

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Now let's understand ABSOLUTE MAXIMUM.

Over the entire domain between 0 and 10 which is a closed interval, the maximum is at A which beats all other peak points in the journey the ball had its bouncy ride. As such, this is also called the global maximum.

There can be only ONE ABSOLUTE MAXIMUM (value on the y axis). However, it is worthy to note that this absolute maximum or minimum can occur at two points on the x axis. Read the local and absolute minimum part below to understand this.

LOCAL MINIMUM AND ABSOLUTE MINIMUM

The LOCAL MINIMUM between a given domain say, 4 and 6 is at C. Look at the graph below:

But over the entire graph, the ABSOLUTE MINIMUM (y value) is at C and D. This means the point at C is not only the local minimum but also an absolute minimum.

Look at the two minimum points on the x axis at C and D, for instance. These are two different values, roughly 5 and 10 (where the green curve is at the lowest). But the absolute minimum is the value on the y axis at these two points where at both points the y value is ONLY 0.

There can be only ONE ABSOLUTE MINIMUM (value on the y axis). However, we saw that this absolute maximum or minimum can occur at two points on the x axis (In this case, 5 and 10).

Mathematically, the general format of the local minimum can be written as follows:

The graph is f(x) and the domain is x1<x<x2. Hence f(a) where 'a' is the value on the x axis at the local minimum C (a=5.5) is less than or equal to all other minimums within this domain.

You can identify 'a' in the graph given above (the yellow line at C).

We got the concept using graphs Donald Trump. Let's get to numbers in the next blog. Let's see how we could use derivatives to find these Extrema (Oh I forgot to tell you. Both minimum and maximum are called Extrema).