This site is being phased out.
Maximum and minimum values of functions
We continue with optimization...
The goal is usually like this :
- maximize the area with fixed perimeter,
- minimize the surface area with fixed volume.
- maximize profit with fixed costs, etc
Mathematically we restate a problem like this as follows:
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 $f(x) = |x|$?
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:
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 $f$. $$L = f(a), \qquad M = f(b)$$
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.
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).
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...
Does it attain its max value? No. (In fact, max is infinite.)
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.