What is the difference between global and local maxima and minima?

[dev70]

hi pf, this is my 1st math question. i am a undergraduate grade 12 student and my question is
what is the basic difference between local maxima minima & global maxima minima?

 

[tiny-tim]

hi dev70!

the top of k2 is a local maximum

the top of mount everest is a global maximum

 

[dev70]

mathematically, what's the difference?

 

[tiny-tim]

tell us what you think, and then we'll comment!

(if necessary, do some research on wikipedia etc )

 

[HassanEE]

If I may interject here. Take a look at this graph http://www.stat.purdue.edu/images/Research/Profiles/velocityReal2.jpg you'll see many humps (maxima). All the humps are local maxima in the sense that they are all peak values at their respective intervals. However, the one at around 0.58 is crowned the global maximum because not only is it the peak in its respective interval, but it is the highest of all the peaks in the entire function.

Take the everest example, you're asked to find the highest peaks in each country around the globe, and call each a local maximum. Then choose the highest peak of all, and call it a global maximum.

The same logic follows for minima.

I hope you found my answer helpful.]

 

[THSMathWhiz]

A global maximum is the absolute greatest value that a function reaches on its domain. For example, the function $f(x)=x^3+x^2-17 x+15$ has no global maximum, but $g(x)=\sin(x)$ has global maxima at $(\frac{\pi}{2}+2\pi n,1),\,\,n\in\mathbb{Z}$. A local maximum is the greatest value that a function reaches within a subset of its domain. For example, the local maximum of $f(x)$ on the set $\{x\colon -5<x<1\}$ occurs at $(-\frac{1}{3}(1+2\sqrt{13}),\frac{1}{27}(560+208\sqrt{13}))$, but $g(x)$ has no local maximum on the set $\{x\colon 0\leq x<\frac{\pi}{2}\}$ (it would have one at the right endpoint, but that is not included in the set).