Local and global maxima

Hi Professor!

Just very quickly, what’s the difference between the global maximum/minimum of a function and the local maximum/minimum of a function? I understand that if a function achieves a global maximum at a point then it automatically achieves a local maximum, but vice versa is not necessarily true.

Also, the definitions look identical besides the local maximum def. containing epsilon.

Cheers!

Reply:

A function f has a local maximum at c if f(c) is \geq all the values at nearby points. f has a global maximum at c if f(c) is \geq the values at all points (in the domain). Therefore, a global maximum is a local maximum, but *not* vice versa. The best way to show the difference would be to draw a picture, but my graphic skills on the computer are very limited. So I’ll just have to write down a formula. Consider

f(x)=x^3-x

You should be able to see that f(x) has a local maximum at c=-1/\sqrt{3}. Near that point, the graph of f(x) is like a hill with apex above that point. What is the global maximum? Of course there is none, because f(x) keeps getting bigger as you move to the right along the x axis (at least starting from the point x=1/\sqrt{3}). One sometimes turns this into a problem with a global maximum by restricting the domain. That is, if we take the same function except with domain only on the closed interval [-100,100], you should be able to see that the global maximum is at x=100.

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