[MUSIC] People have the idea that a local 
minimum means the function decreases and 
then increases. 
Here's a local minimum on the graph of 
this random function. 
And the misconception is that they all 
look like this. 
That every time you got a local minimum 
on one side, the function's decreasing, 
and on the other side the function's 
increasing. 
Plenty of local minima do look exactly 
like that. But, there's also plenty of 
pathological examples. 
For instance, consider this a somewhat 
pathological example. 
I'm going to define this function f as a 
piecewise function. 
If the input's nonzero, I'm going to do 
this. 1+sin(1/x), which makes sense since 
x isn't zero, time x^2. 
And if the input is zero, the function's 
output will also be zero. 
In this case, there's a local minimum at 
zero. 
How do I know? Well, here's how I know. 
Let's take a look at this function The 
claim is that f(x) is never negative. 
How do I know that? Well, what do I know 
about sine? Sine of absolutely anything 
at all, no matter what I take this sine 
of, is between -1 and 1. 
Now, if I add 1 to this, 1 plus sine of 
absolutely anything at all, 
is between zero and two. 
Now, that's pretty good. 
Now, think back to the definition of this 
function. 
Here, I've got 1 plus sine of something, 
it doesn't matter what, 
alright? 
1 plus sine of anything, 
this is between zero and two. 
Now, 
I'm multiplying it by x^2. 
What do I know about x^2? Well, x^2 is 
not negative. 
It could be zero, it could be positive. 
But no matter what x is, x^2 is not 
negative. 
Now, I'm multiplying 1+sin(1/x), this 
number which is trapped between zero and 
to, by x^2 which is never negative. And 
that means f(x) is not negative as long 
as x isn't equal to zero, 
alright? 
As long as x isn't equal to zero. I mean, 
this first case and this is a 
non-negative number times a non-negative 
number, so the product is also 
non-negative. 
Now, the other possibility, of course, is 
that I plug in zero for x. 
But then, f(0) is just by definition 
zero. 
And that means in either case, no matter 
that I plug in for x, f(x) is never a 
negative. 
Now if f(x) is never negative and f(0)=0, 
then I know that this must be the 
smallest possible output value for the 
function. 
The only numbers that are smaller than 
zero are negative numbers, and the output 
of this function is never negative. But 
this isn't the usual sort of local 
minimum where the function just decreases 
and then increases. 
Well, here's the graph for our funciton 
f. 
And, there is a local minimum at zero, 
but if I start zooming in, no matter how 
much I zoom in, there's no little region 
on which the graph is just decreasing and 
then increasing. 
The graph is always wiggling. 
The upshot here is that decreasing and 
then increasing is one way to produce a 
local minimum, but it's not the 
definition of a local minimum. 
And not every local minimum arises in 
that exact way. 
What a local minimum means is just that 
no nearby output value is smaller than 
that local minimum value.