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[MUSIC] People have the idea that a local 
minimum means the function decreases and 

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then increases. 
Here's a local minimum on the graph of 

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this random function. 
And the misconception is that they all 

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look like this. 
That every time you got a local minimum 

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on one side, the function's decreasing, 
and on the other side the function's 

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increasing. 
Plenty of local minima do look exactly 

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like that. But, there's also plenty of 
pathological examples. 

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For instance, consider this a somewhat 
pathological example. 

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I'm going to define this function f as a 
piecewise function. 

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If the input's nonzero, I'm going to do 
this. 1+sin(1/x), which makes sense since 

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x isn't zero, time x^2. 
And if the input is zero, the function's 

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output will also be zero. 
In this case, there's a local minimum at 

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zero. 
How do I know? Well, here's how I know. 

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Let's take a look at this function The 
claim is that f(x) is never negative. 

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How do I know that? Well, what do I know 
about sine? Sine of absolutely anything 

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at all, no matter what I take this sine 
of, is between -1 and 1. 

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Now, if I add 1 to this, 1 plus sine of 
absolutely anything at all, 

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is between zero and two. 
Now, that's pretty good. 

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Now, think back to the definition of this 
function. 

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Here, I've got 1 plus sine of something, 
it doesn't matter what, 

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alright? 
1 plus sine of anything, 

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this is between zero and two. 
Now, 

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I'm multiplying it by x^2. 
What do I know about x^2? Well, x^2 is 

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not negative. 
It could be zero, it could be positive. 

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But no matter what x is, x^2 is not 
negative. 

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Now, I'm multiplying 1+sin(1/x), this 
number which is trapped between zero and 

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to, by x^2 which is never negative. And 
that means f(x) is not negative as long 

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as x isn't equal to zero, 
alright? 

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As long as x isn't equal to zero. I mean, 
this first case and this is a 

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non-negative number times a non-negative 
number, so the product is also 

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non-negative. 
Now, the other possibility, of course, is 

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that I plug in zero for x. 
But then, f(0) is just by definition 

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zero. 
And that means in either case, no matter 

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that I plug in for x, f(x) is never a 
negative. 

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Now if f(x) is never negative and f(0)=0, 
then I know that this must be the 

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smallest possible output value for the 
function. 

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The only numbers that are smaller than 
zero are negative numbers, and the output 

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of this function is never negative. But 
this isn't the usual sort of local 

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minimum where the function just decreases 
and then increases. 

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Well, here's the graph for our funciton 
f. 

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And, there is a local minimum at zero, 
but if I start zooming in, no matter how 

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much I zoom in, there's no little region 
on which the graph is just decreasing and 

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then increasing. 
The graph is always wiggling. 

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The upshot here is that decreasing and 
then increasing is one way to produce a 

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local minimum, but it's not the 
definition of a local minimum. 

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And not every local minimum arises in 
that exact way. 

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What a local minimum means is just that 
no nearby output value is smaller than 

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that local minimum value.