So, let me hit you with a theorem,
right straight from his beginnings of [inaudible].
In case of convex functions,
global extremums are local extremums
and local extremums are global extremums they coincide.
So why does it happen?
It's actually quite nice to
understand the easy and intuitive.
Assumes that you have some convex function for
example x squared but no pressure here,
and you have drawn
several tangent lines at
several points like I did on this graph.
What do you see?
The question here is,
what is the slope of these curves,
and how it changes
throughout going from the left to the right?
As you see, this is the lowest negative value.
This is a bit higher negative value.
This is a positive value,
is this is hypo-style value.
So, the idea here is that the slope actually
rises from the left part to
the right part from left to right.
Our doing what you've actually increases from
negative to positive values.
If function Z, starts from
the negative values and
goes to the positive values at the very end.
At some point it necessarily shoot across 0,
right and if it crosses 0 there is an extremum.
So for example,
set our connection with basic extremum theorems.
Let us have a look at what we've accomplished here.
First of all, we don't understand how the derivative
actually connected whereas in
convexity or concavity here.
If the derivative of
function changes monotonically, monotonically rises.
Then the second derivative is positive.
Then function look like x squared.
Well and also rule here,
if the second derivative is positive, positive is good,
then there is a nice smile here and your
happier and happy smile looks like
concave function and the derivative is negative.
As the first derivative is falling,
falls the segment, then the function is concave.
Well in other words negative derivity is bad sign and
bad smile looks like this
and since bad smile
look like this, that's concave function.
Also we understand the last thing.
The last thing that seems
where monotonically changing as we adjust
the circuit and we have a 0 since we are
crossing from positive to
negative or from negative to positive case.
Then we have only 1 point
where our derivative equals
to zeros as we have only one extremum.
That's actually quite the case for our theorem.
We've just proved it.
You haven't noticed.
We proved the case that in case we have
a convex or concave functions,
our global extremum actually
is local extremum and vice versa.
There is only one of them. That is fun.