What is the significance of the global maximum in differentiable functions?

In summary, we are trying to show that for a differentiable and continuous function with a global maximum at , the following holds: To prove this, we use the definition of the derivative at $
  • #1
mathmari
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Hey! :eek:

I am looking at the following:

The differentiable and so also continuous function gets its global maximum at a point . Show that the following holds

Could you give me a hint what we are supposed to do? (Wondering)

I have done the following:

Since the function has its global maximum at we have that .

We use the definition of the derivative at :
For we have the following:

Let . We have the interval . , and so is well defined for .

Then we have:

For we have the following:

Let. We have the interval . , and so is well defined for . So, it holds that .

Then we have:
For we have the following:


Is everything correct? Could I improve something? Are the one-sided limits correct? (Wondering)
 
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  • #2


Hi there! It looks like you're on the right track. To show that , you can use the fact that is differentiable at and the definition of the derivative. Specifically, you can show that the one-sided limits of the derivative at are both equal to 0, which would imply that . This would cover all three cases: , , and .

One thing to note is that in your proof for , you have a slight error in your limit calculation. It should be:



Other than that, everything looks good! Keep up the good work.
 

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