Mean Value Theorem II

[Music] now this is part 2 of a two-part series on the mean value theorem in which we talk about applications of the mean value theorem something more substantial than catching speeding drivers we do things like check the conditions for a function to be constant to be increasing to be decreasing if you haven't seen the

first part go back and look at that video otherwise just quickly review the statement of the mean value theorem which is behind me let's first look at how used derivatives to check for a function being constant you know on the one hand that a constant function has derivatives zero for example if you take

constant value five for a function the derivative of five is zero the derivative of a constant is zero what about the other way around if the derivative is zero does that mean that the function has to be constant now that's almost true but here's the kind of thing you have to be careful of this

function has value to 4x negative it has value 3 for X positive and I'm showing you the graph it's not a constant function its domain is all the non zero x values and at every point well every point where the function is defined the derivative is 0 and yet the function is not a constant well okay so that's kind

of a cheat and I agree and this is really the only kind of exception that can happen if if the function is to find on an interval you don't have this kind of behavior so what's special about intervals is they're connected you can get from any point to any other point inside the domain so domains which are

not connected you can't make any conclusion like this so if your function is defined on an interval and the derivative is 0 then the function does have to be constant on that domain on the interval and I'm just going to talk about an open interval here because it's Manek to talk about derivatives at