Good afternoon students, today we will study maxima and minima increasing decreasing

functions and other applications of Derivatives. Let us start.

Derivatives are very widely used and they have many different applications. As we studied in

the last lecture, firstly they are used in finding equation of tangents, equation of normals.

Secondly, they are also used for checking whether the function is increasing or decreasing.

Their third application is maxima and minima and mean value of theorems and many other

different applications. We have talked about tangents and normals in last class, today we

will talk about increasing, decreasing functions, maxima and minima and mean value

theorems.

Function is increasing when derivative is positive. Function is decreasing when derivative is

negative. When derivative is positive function is increasing function and when derivative is

negative function is decreasing function. If derivative is positive function will increase and

derivative is negative function will decrease. Increasing function will look like this, going

from left to right, value of function increases, graph goes upwards and decreasing function is

like this, going from left to right and graph goes downwards. Decreasing function and

increasing function. This graph is made when derivative is negative and this graph is made

when derivative is positive.

Maxima and Minima – I hope everybody had studied maxima and minima for school, yes,

very good. This is exactly same as what you studied for boards. Exactly, same types of

questions are asked. Firstly we have to find critical points. Critical points are the points

where f dash x is zero or not defined. First of all we will find critical points and after that we

will check if critical point is maxima or minima. Second point is we check whether critical

point is maxima or minima. Critical point is that point where derivative is zero. Now, how to

check whether critical point is maxima point or minima point. Say, x equals to a is critical

point, now we find f double dash a. We will find double derivative, if it is negative then it is

maxima or f double dash a is positive then x equals to a is minima. I hope everybody got it.

Yes, very good, to check maxima and minima of functions, we find critical points. Critical

points are the points whether f dash is zero or not defined. After that we can apply two ways

firstly what I have taught you is second derivative test. You find double derivative and put

critical point if the answer is negative then critical point is maxima point or if it is positive

then it is minima point. But double derivative point many a times fail, when double

derivative doesn’t exists only or double derivative instead of being positive or negative is

zero.

So, we have one more test which is first derivative test. What do first derivative test says? It

says if sign of derivative changes from negative to positive, what is this – sign of f dash x, on

the both sides of critical point, we have to check the sign of f dash x, if it goes from negative

to positive, then critical point is, function has increased instead of decreasing. As soon as the

function increases instead of decreasing, critical point is point of minima. You can very

clearly say function is increasing instead of decreasing. Means the value decrease at a

minimum value and then increases, so the critical point is of minima. And if sign of f dash

becomes turns positive to negative on the both sides of critical point. Means f dash should

change from positive to negative, after the functions increasing it decreases so the critical

point is point of maxima. To find maxima and minima functions, we need to do two things,

first is we need to critical points, second thing is we should check whether critical point is of

maxima or minima, for this we have two tests. Second derivative test and first derivative

test. In second derivative test, in double derivative value we will put critical value and if it is

positive then minima and if it is negative then maxima. In first derivative test, on both the

sides of critical points, we will check sign of f dash, if first derivative sign changes from

negative to positive, so the function is from decreasing to increasing, means this point will

be of minima. And if first derivative sign changes from positive to negative, then the function

will go from increasing to decreasing. So this point will be of maxima. And third thing which

may happen is.