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.