Question

Let f : [a,b] → R be a bounded function and let:             M = sup...

Let f : [a,b] → R be a bounded function and let:

            M = sup f(x)
            m = inf f(x)
            M* =sup |f(x)|
            m* =inf |f(x)|

assuming you are taking values of x that lie in [a,b].

Is it true that M* - m* ≤ M - m ?

If it is true, prove it. If it is false, find a counter example.

Math   Advanced-Math   
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Homework Answers

Answer #1

The result is true:

Proof:

CASE-1:

Suppose

Then

Hence

Then .

CASE-2:

Again Suppose that such that and such that  .

then we must have   which in turn implies that

If then by case (1) the result is true.

If then

which in turn implies that

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