Extreme Value Theorem

Question

Please explain The Extreme Value Theorem. Specifically explain how a jump discontinuity and an infinite discontinuity will prevent a maximum/minimum in their own unique way.

Assuming the function is continuous, describe the shape of potential extrema where the derivative is undefined.

Also, for a continuous function, describe the shape where the derivative is undefined.

In what scenario will a relative extrema, located at a critical number, not be an absolute extrema on an interval?

Answer 1

If a function is continuous in the closed interval [a, b], then there must be a maximum and minimum extrema in that interval.

 

In a jump discontinuity, a single x value will have 2 different f(x) values.  Do you remember limits?  If the limit approaching the left and right of the same x value is not the same, and f(x) is not defined at that x value, then it is not continuous.

 

An infinite discontinuity means that you have a vertical asymptote at the x value in the interval.  The limit approaching from one side is infinity, where the limit approaching from the other side is negative infinity.

 

An example of a shape of a potential extrema is a concave up and concave down shape.

 

Concave up - increases slowly, then increases at fast rate later on.  The endpoint of the fast rate creates a vertical line

Concave down - decreases slowly, then decreases slowly later on.

 

Exponential functions such as growth and decay.  If you graph this type of function, it appears to have an extrema at its turning point.  However, it has no extrema because the graph either only decreases or only increases.  In order to have an extrema, there must be increases and decreases periodically.

 

 

SCENERIO:

 

cp is the critical number.   Under these conditions:

 

For maximum

 

f(a) > f(cp)             or           f(b) > f(cp)

 

 

For minimum

 

f(a) < f(cp)            or            f(b) < f(cp)

 

 

In other words, f(cp) cannot be greater than or less than f(x) at the end boundaries.