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[MUSIC] The goal of computing is not 
numbers but insight. 

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I don't care that f(2) is equal to four. 
I do care about the qualitative features 

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of a function. 
Here's a graph of some random function I 

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just made up. 
A significant qualitative feature of this 

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graph is that right here is a valley in 
the graph and up here is a mountaintop. 

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Speaking of mountaintops and valleys is 
maybe too metaphorical, not precise 

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enough. 
Let's try to work out a better 

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definition. 
So, instead of calling this a mountain 

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top, I'm going to call that a local 
maximum. 

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Let's be even a little bit more precise. 
Let's suppose that that maximum value 

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occurs at the input c, where the output 
is f(c). 

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I'm going to call f(c) the local maximum 
value of the function near the input c. 

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Here's a precise definition, 
f(c) is a local maximum value for f if 

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whenever x is near the input c, f(c) is 
bigger than or equal to f(x). 

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Maybe this isn't even precise enough. 
A big sticking point with this is this 

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word near. 
That's sort of a weasley word. 

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I can make this near precise just like 
we've been doing with limits. 

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I'll introduce sum epsilon. 
So f(c) is a local maximum value for the 

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function f, if there's some small number, 
that's what I mean by near. 

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So that whenever x is near c, 
and by near c, now, I'm in between c- 

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epislon and c+ epsilon. 
This is really close to c if epsilon is 

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real small. 
Okay. 

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So that, whenever x is contained in this 
interval, f(c) is bigger then or equal to 

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f(x). 
We can give a similar sort of definition 

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for the valleys. 
Here's that same graph again and I've 

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highlighted a local minimum on the graph 
of this function. 

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Near the input c, f(c) is the smallest 
output for the function. 

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A little bit more precisely, I'm calling 
it a local minimum value, 

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because whenever x is near c, f(c) is 
less than or equal to f(x). 

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Or even a little bit more precisely again 
just like for local maximums, I can 

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replace near with espilon. 
So f(c) is a local minimum value for the 

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function, if there's some epsilon 
measuring the nearness, so that whenever 

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x is between c- epsilon, and c+ epsilon, 
f(c) is less than or equal to f(x). 

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Sometimes, I'm going to want to talk 
simultaneously about local maximums and 

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local minimums. 
So, we'll call either a local minimum or 

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a local maximum a local extremum. 
And this is kind of a pretentious word, 

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but it's just a word that we can use so 
that we can talk about both of these 

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concepts simultaneously, 
because they actually share quite a few 

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features in common. 
What if I want to talk about multiple 

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extremums? Well, what if we wanted to 
talk about an octopus? Well, that's not 

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really a problem, 
but, what if you wanted to talk about two 

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of these things? You might not call it an 
octopus, you might call it an octopuses 

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now. 
But, you'll find some people will get 

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angry at you if you do that and I want 
you to call these octopi. 

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I probably don't really agree with them, 
but, the same problem comes up with 

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minimums and maximums and extremums. 
You're going to find some people who will 

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demand that you call these minima, 
maxima, and extrema if you're talking 

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about more than one of these things, 
but really, either is fine. 

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The world local here is really in 
contrast to the world global. 

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We'll say, in fact, everyone will say 
that f(c) is a global maximum value for 

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the function f if no output of the 
function is larger than f(c). 

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Maybe that's too cute of a way to say it. 
Here's a more precise way to say it, 

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f(c) is a global maximum value for the 
function if whenever x is in the domain 

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of f, 
I'm only going to be considering inputs 

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in the domain, of course, 
then f(c) is bigger than or equal to 

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f(x). 
Do the same deal for minimal values, 

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f(c) is a global minimum value for the 
function if whenever I've got a point in 

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the domain f(c) is less than or equal to 
f(x). 

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One subtle thing to point out here is 
that I'm not claiming, say for global 

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maximum values that this is the biggest 
output of the function. 

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What I'm saying is that any other output 
isn't larger than f(c), 

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f(c) is bigger than or equal to any other 
output of the function and the same deal 

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for this global minimum. 
I'm not saying this is the smallest 

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output of the function, I'm just saying 
that any other output is bigger than or 

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equal to this output. 
Now we can see some examples of this. 

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Of this. 
Here's that same graph we've been looking 

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at so many times here. 
Let's try to figure out where the local 

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extrema are. 
this point here is a local maximum, 

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that's the biggest output value for 
nearby input values. 

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This point here is a local minimum, 
right? You're sitting in that valley, 

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that's the smallest output valley among 
nearby inputs. And, up here is another 

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local max and this local maximum is also 
a global maximum. 

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I mean, if you're really assuming that 
this graph just continues down to the 

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left and the right-hand sides, 
you should also note that if you really 

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believe this thing continues down like 
this, this function has no global 

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minimum. 
Nobody's promising you that there is a 

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global minimum or a global maximum, but 
in this case, there isn't one. 

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There's no global minimum. 
There can definitely be multiple local 

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maximums, 
but there can also be multiple global 

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maximums. 
So here, I've drawn another graph, where 

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I've rigged it so that these two output 
values are the same. 

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So these are both local maximums, 
but they're also both global maximums. 

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Alright? So in this case, these are both 
global and also local maximums. 

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There's even more, dare I say it, extreme 
version of this. 

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Here, I've graphed the constant function 
y=17 is both a global maximum and a 

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global minimum for this constant 
function. 

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The distinction between local and global 
maximums is really quite important, even 

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in everyday life. 
When you're standing at a local maximum, 

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on the top of this mountain, small 
changes to your situation just make 

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things worse, 
and yet, if you're willing to go through 

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this valley, you'll eventually come up 
here to what at least appears to be to 

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global maximum of this function. 

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[MUSIC]