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Maximum and minimum values of functions

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We continue with optimization...

Fixed Volume.

The goal is usually like this :

Mathematically we restate a problem like this as follows:

Minimum and Maximum Points on a Graph.

How can we use the derivative here?

Observe: these special points have horizontal tangents, so $f^{\prime}(a) = 0$.

Easy to find these, but what about $f(x) = |x|$?

$. $f^{\prime}(0)$ does not exist.

The answer is: put those on the list as well.

These points are called critical points. They form a (short) list of candidates for max and min.

More formally: A function $f$ has a global (absolute) maximum at $x = a$ if $f(x) \leq f(a)$ for all $x$ in the domain of $f$. Then $a$ is called a global max point. $f(a)$ is called the global max value.

Max points with the same max value; which is unique:

Graph showing the same max points.

Max and min points are also called extreme points/values.

A function $f$ has a global (absolute) minimum at $x= a$ if $f(x) \geq f(a)$ for all $x$ in the domain of $f$. Then

  • $a$ is called a global min point,
  • $f(a)$ is called the global min value.

In this case the min/max value, $y$, is attained by $f$. $$L = f(a), \qquad M = f(b)$$

Min/Max values from $f$.

Example of this : $\sin x \leq 1$, 2 not attained.

On the picture, $f$ captures ("attains") every value between $f(a)$, $f(b)$. That's every intermediate value.

IntermediateValueTheorem.png

More precisely...

Intermediate Value Theorem. Given a continuous function $f$ $[a,b]$, for each $L$ in $[f(a), f(b)]$ (or $[f(b), f(a)]$), there is $c$ in $[a,b]$ such that $f(c) = L$.

Further, on the next picture $f$ attains its extreme values (max and min).

ExtremeValueTheorem.png

Extreme Value Theorem. If $f$ is continuous on $[a,b]$, then there is $c,d$ in $[a,b]$ such that $f(c) = M,f(d) = L$, where $M,L$ are the maximum and minimum values of $f$ respectively.

Why are these conditions important?

Consider these examples...

Graph of $y=\frac{1}{x}$

Does it attain its max value? No. (In fact, max is infinite.)

$M$ is not attained.

Here max is not attained again. Indeed there is no $c: f(c) = M$.

But why does theorem not apply?

Because $f$ is not continuous.

Lesson: always check conditions of a theorem before you apply it.

Example: Global Max/Min for Constant Function

For a constant function, all $x$'s are both global max and min points.


Q: Why do we need EVT?

A: EVT implies that there is a solution for the optimization problem.