CBSE Class 12 > Class 12 Videos > Increasing, decreasing function - Application of Derivatives
Increasing, decreasing function - Application of Derivatives Video Lecture
Text Transcript from Video
hello friends this video an application
of derivatives part 8 is brought to you
by exam Viacom no more fear from exam
before watching this video please make
sure that you have watched part 1 to
part 7 tickets take 1 theorem and before
we take this theorem let me tell you why
are we taking the student the reason why
we are taking this theorem is the
current approach to find whether the
function is increasing or decreasing is
we have to first find f of X 1 then we
have to find f of X 2 current and then
it may be true that the point which we
are selected for this this is true but
this may not be true for all the point X
mixed correct so in that case I'll show
you an example for example if you take
this graph correctly and my graph is
like this correct this this this goes
down as colossal like this it will draw
once again my graph is like this if I
take my point X 1 this and X 2 is this
guy correct
so with the formula I know that this guy
is greater than this very is greater
than this guy so with my current formula
I'll be able to prove that see the
current formula I can see that X 2 is
greater than X 1 correct and f of X 2 is
also greater than X why this guy is f of
X 2 this guy is f of X 1 so this guy is
greater f of X 2 is greater than f of X
1 this is true excuse me to the next -
history but we know that this guy is not
increasing all right so logically what
we have do is we can't tell the function
is increasing when it is just based on
two points we have to keep on finding
other points also whether correct so
this is the problem because if you take
a long
different in external x2 you may find
the solution we say that as function is
increasing or decreasing but actually it
is not increasing or do you see to avoid
such kind of thing we have a new formula
that says that if F is a continuous
function on a and B and differentiable
on the open interval a and B then F is
increasing in this interval if F dash X
is greater than 0 for each X that is
member of a coma also f dash is a
decreasing function if F dash X is less
than 0 for each member of 4x member of a
comma B for each please note for each
here also called each is there because
for each and every value of x here I
have to show that also f is a constant
function if F dash X is 0 for each value
of xn so with this and you note find the
value of F of x1 f of x2 everywhere I
can just find the value of F dash X
current and these X have to find for all
in this range so the term which makes
our life simple to find where the
function is increasing or decreasing
function the only example we used to
find f of x1 f of x2 and still they were
loopholes there correct because we have
to find all x1 x2 here we just have one
value X and then for example is X now at
this X I want to know where the function
is increasing or decreasing I can find
using this because this formula is
independent of X 1 and X 2 at that
particular point we can find where the
function is increasing what is the early
formula we had two inputs x1 and x2 and
sometimes the results wait I see also
now with this I have only one point I
have to tell whether at this point the
function is increasing or decreasing so
the formula you do not have to do
anything you have to just differentiate
the function if find F dash X correct so
if you have this function you will find
f dash X now if this is less than zero
this better than there are two options
it is greater than zero it is increasing
it is less than zero it is decreasing
now you may be wondering how I got this
formula so let's think through of this
theorem to prove this theorem we'll use
the same logic that if X 1 is less than
X 2 any numbers in the domain and if my
f of X 2 is greater than f of X 1 then
my function is increasing function and
if for in his in punching and if my f of
X 1 is greater than f of X 2 then our
function is decreasing possible the same
logic will use so let us assume let X 1
and X 2 be two values a comma B because
I have my function I am my function f of
X whose domain is a comma B now if I
want to find f - C or a des that any
pointed suppose I define f - this is
nothing but f of X 2 minus f of X 1 by X
2 - correct
where C is a number between X 1 X 2 if
you see how I have done this if this is
my graph correct if my X 1 is this guy
x2 this guy I want to find a slope at
this point it was C so this is nothing
but slope we know that that is slope and
this guy when I say F to F x2 minus F x1
there is nothing but y2 minus y1 because
these guy if you take this point these
point is nothing but X 2 comma f of X 2
correct and this point is nothing but ax
right here X 1 comma f of X F correct so
this is nothing but y 2 minus y 1 by X 2
minus X now this guy is slope so slope
at this of this line is nothing but F
dash C and that is nothing where Y 2
minus y 1 by X minus 1 X 1 y 2 and y 1
is nothing but f of X 2 and f of X 1 so
this is what I get now if this guy is
positive this guy is positive case 1
this guy is positive if this guy is
positive see X 2 minus X 1 is always
positive correct because X 2 is greater
than X 1 this guy is always positive
always positive if this guy is positive
F dash C is positive so case 1 F dash C
is greater than 0 in that case f of X 2
minus or f of X 2 is greater than f of X
1 correct
why because f of X 2 minus f of X 1 has
to be greater than 0 or this implies f
of X 2 is greater than f of X in case 2
when I say that F dash C is less than 0
so this is positive this has to be less
than 0 so in that case f dot this minus
f of X 1 this is less than 0 or I can
say that f of X 2 is less than never
this implies if you see this implies
this guy Stevens's that means function
is decreasing and here this is nothing
but this guy this implies function is
increasing correct so if F Dash is
greater than zero function is increasing
if F Dash is less than zero function is
decreasing Casely miies F dash C is
equal to zero that is the case since the
denominator is always 0 or a positive in
that case f of X 2 minus f of X 1 has to
be funny in that case f of X 2 is equal
to f of X if that is true that means
function is constant like this putting
any value X 2 X 1 this guy is safe this
guy is f of X 2 this guy is f of X this
is X 1 X 2 correct so what we have done
we have proved this theorem and theorem
say if F dash C is greater than 0 the
function is increasing at that point C
if F dash C is less than 0 the function
is decreasing at the Point C if F dash C
is equal to 0 the function is constant
at that point we'll take some examples
of increasing and decreasing function so
you have to prove that FX 3x + 17 is
strictly increasing on out so we have
done similar kind of motion where we
have taken 2 points X 2 and X 1 and then
we have found X 2 and f of X 2 and f of
X 1 and we showed that f of X 2 was
greater than f of X 1 for X 2 is greater
than X 1 right so we have done this kind
of motion where we have taken X 2 is
greater than X 1 and we have proved that
f of X 2 is all this
now we'll do with the easier way the
easier way see is that if F dash of X is
greater than zero for each X that is
part of this domain that is real number
then I can say that is always greater is
not good it is good if f dash X is
greater than zero for each X then I can
see that the function is strictly using
Europe so let's find a plus X so if my f
of X is 3x plus 17 my f dash X will be
nothing but click and this guy is all
listed you take any value of X is
greater than zero why it is independent
of term exported so this guy is greater
than zero for any X that is no so I can
say that this function is strictly equal
Z Y strictly because it is greater than
0 it is not greater than equal to is
greater than zero correct let us take
one more example again we have this
function e to the power 2 X we have to
prove that this is strictly increasing
or not same thing we have to find
through that F dash X is always greater
than 0 for each X that is member of this
domain that is please note for each X so
let's find em des X F dash X is nothing
but e to the power 2 X into 2 now we
know that e to the power something is
always greater than 0 is always positive
and positive into positive is always
positive this guy's always greater than
0 for any X that is member you take any
value of x
they give you even if you take negative
value of x this guy is post e to the
power X is the reversible minus 5 minus
5 into is minus 10 e to power minus 10
is also for the negative value of x this
guy's positive so this is strictly
increasing function thank you visit exam
viacom to watch free educational videos
try free online tests get the best
quality study materials study from the
best tutors and mentors and much more
thanks again
of derivatives part 8 is brought to you
by exam Viacom no more fear from exam
before watching this video please make
sure that you have watched part 1 to
part 7 tickets take 1 theorem and before
we take this theorem let me tell you why
are we taking the student the reason why
we are taking this theorem is the
current approach to find whether the
function is increasing or decreasing is
we have to first find f of X 1 then we
have to find f of X 2 current and then
it may be true that the point which we
are selected for this this is true but
this may not be true for all the point X
mixed correct so in that case I'll show
you an example for example if you take
this graph correctly and my graph is
like this correct this this this goes
down as colossal like this it will draw
once again my graph is like this if I
take my point X 1 this and X 2 is this
guy correct
so with the formula I know that this guy
is greater than this very is greater
than this guy so with my current formula
I'll be able to prove that see the
current formula I can see that X 2 is
greater than X 1 correct and f of X 2 is
also greater than X why this guy is f of
X 2 this guy is f of X 1 so this guy is
greater f of X 2 is greater than f of X
1 this is true excuse me to the next -
history but we know that this guy is not
increasing all right so logically what
we have do is we can't tell the function
is increasing when it is just based on
two points we have to keep on finding
other points also whether correct so
this is the problem because if you take
a long
different in external x2 you may find
the solution we say that as function is
increasing or decreasing but actually it
is not increasing or do you see to avoid
such kind of thing we have a new formula
that says that if F is a continuous
function on a and B and differentiable
on the open interval a and B then F is
increasing in this interval if F dash X
is greater than 0 for each X that is
member of a coma also f dash is a
decreasing function if F dash X is less
than 0 for each member of 4x member of a
comma B for each please note for each
here also called each is there because
for each and every value of x here I
have to show that also f is a constant
function if F dash X is 0 for each value
of xn so with this and you note find the
value of F of x1 f of x2 everywhere I
can just find the value of F dash X
current and these X have to find for all
in this range so the term which makes
our life simple to find where the
function is increasing or decreasing
function the only example we used to
find f of x1 f of x2 and still they were
loopholes there correct because we have
to find all x1 x2 here we just have one
value X and then for example is X now at
this X I want to know where the function
is increasing or decreasing I can find
using this because this formula is
independent of X 1 and X 2 at that
particular point we can find where the
function is increasing what is the early
formula we had two inputs x1 and x2 and
sometimes the results wait I see also
now with this I have only one point I
have to tell whether at this point the
function is increasing or decreasing so
the formula you do not have to do
anything you have to just differentiate
the function if find F dash X correct so
if you have this function you will find
f dash X now if this is less than zero
this better than there are two options
it is greater than zero it is increasing
it is less than zero it is decreasing
now you may be wondering how I got this
formula so let's think through of this
theorem to prove this theorem we'll use
the same logic that if X 1 is less than
X 2 any numbers in the domain and if my
f of X 2 is greater than f of X 1 then
my function is increasing function and
if for in his in punching and if my f of
X 1 is greater than f of X 2 then our
function is decreasing possible the same
logic will use so let us assume let X 1
and X 2 be two values a comma B because
I have my function I am my function f of
X whose domain is a comma B now if I
want to find f - C or a des that any
pointed suppose I define f - this is
nothing but f of X 2 minus f of X 1 by X
2 - correct
where C is a number between X 1 X 2 if
you see how I have done this if this is
my graph correct if my X 1 is this guy
x2 this guy I want to find a slope at
this point it was C so this is nothing
but slope we know that that is slope and
this guy when I say F to F x2 minus F x1
there is nothing but y2 minus y1 because
these guy if you take this point these
point is nothing but X 2 comma f of X 2
correct and this point is nothing but ax
right here X 1 comma f of X F correct so
this is nothing but y 2 minus y 1 by X 2
minus X now this guy is slope so slope
at this of this line is nothing but F
dash C and that is nothing where Y 2
minus y 1 by X minus 1 X 1 y 2 and y 1
is nothing but f of X 2 and f of X 1 so
this is what I get now if this guy is
positive this guy is positive case 1
this guy is positive if this guy is
positive see X 2 minus X 1 is always
positive correct because X 2 is greater
than X 1 this guy is always positive
always positive if this guy is positive
F dash C is positive so case 1 F dash C
is greater than 0 in that case f of X 2
minus or f of X 2 is greater than f of X
1 correct
why because f of X 2 minus f of X 1 has
to be greater than 0 or this implies f
of X 2 is greater than f of X in case 2
when I say that F dash C is less than 0
so this is positive this has to be less
than 0 so in that case f dot this minus
f of X 1 this is less than 0 or I can
say that f of X 2 is less than never
this implies if you see this implies
this guy Stevens's that means function
is decreasing and here this is nothing
but this guy this implies function is
increasing correct so if F Dash is
greater than zero function is increasing
if F Dash is less than zero function is
decreasing Casely miies F dash C is
equal to zero that is the case since the
denominator is always 0 or a positive in
that case f of X 2 minus f of X 1 has to
be funny in that case f of X 2 is equal
to f of X if that is true that means
function is constant like this putting
any value X 2 X 1 this guy is safe this
guy is f of X 2 this guy is f of X this
is X 1 X 2 correct so what we have done
we have proved this theorem and theorem
say if F dash C is greater than 0 the
function is increasing at that point C
if F dash C is less than 0 the function
is decreasing at the Point C if F dash C
is equal to 0 the function is constant
at that point we'll take some examples
of increasing and decreasing function so
you have to prove that FX 3x + 17 is
strictly increasing on out so we have
done similar kind of motion where we
have taken 2 points X 2 and X 1 and then
we have found X 2 and f of X 2 and f of
X 1 and we showed that f of X 2 was
greater than f of X 1 for X 2 is greater
than X 1 right so we have done this kind
of motion where we have taken X 2 is
greater than X 1 and we have proved that
f of X 2 is all this
now we'll do with the easier way the
easier way see is that if F dash of X is
greater than zero for each X that is
part of this domain that is real number
then I can say that is always greater is
not good it is good if f dash X is
greater than zero for each X then I can
see that the function is strictly using
Europe so let's find a plus X so if my f
of X is 3x plus 17 my f dash X will be
nothing but click and this guy is all
listed you take any value of X is
greater than zero why it is independent
of term exported so this guy is greater
than zero for any X that is no so I can
say that this function is strictly equal
Z Y strictly because it is greater than
0 it is not greater than equal to is
greater than zero correct let us take
one more example again we have this
function e to the power 2 X we have to
prove that this is strictly increasing
or not same thing we have to find
through that F dash X is always greater
than 0 for each X that is member of this
domain that is please note for each X so
let's find em des X F dash X is nothing
but e to the power 2 X into 2 now we
know that e to the power something is
always greater than 0 is always positive
and positive into positive is always
positive this guy's always greater than
0 for any X that is member you take any
value of x
they give you even if you take negative
value of x this guy is post e to the
power X is the reversible minus 5 minus
5 into is minus 10 e to power minus 10
is also for the negative value of x this
guy's positive so this is strictly
increasing function thank you visit exam
viacom to watch free educational videos
try free online tests get the best
quality study materials study from the
best tutors and mentors and much more
thanks again
About this Video
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Apr 22, 2026 Last updated
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