L21 : First Derivative Test - Application of Derivatives, Maths, Class 12 Video Lecture

hello friends this video an application
of derivatives part 21 is brought to you
by exam feel calm no more free from exam
before watching this video please make
sure that you have watched part 1 to bar
20 for example using first derivative
test we had to find the local maxima
minima for this function so if you see
this function we know that this is open
interval function and have to find local
maximum is as I told for open interval
function the local maxima minima can
happen at FX x equal to 0 or non
differentiable points since it is a
continuous function the non
differentiable point does not come
because it has all the points definition
so only thing is had to find all the
points where F does X is equal to 0
critical points so critical points are
points where F dash X equal to 0 so let
us find f dash x f dash x is nothing but
x cube becomes 3 x squared - 3 x becomes
3 and 3 become 0 that is 3 into X square
minus 1 that is 3 into X plus 1 into X
minus 1 and this is equal to 0 so I get
X is equal to 1 or minus 1 so either one
of these Maxima or minima you don't know
so to find whether it is a maximum or a
minimal it's do one thing let's have the
graph like this like this is minus 1 and
this is 1 correct and if we have to find
whether at these points my it is maximum
or minimum so let us take 1 first so if
you take 1 I know my slope at 1 is C
that I know so slope at this guy's so it
is something like this slope y because F
of 1 is let us take a number just next
to this that is F of 1 point 1 that guy
is nothing but 3 into 1 point 1 square
- 1 that is 3 into 1 point 2 1 minus 1
that is 3 into point Q 1 that is 0.63 I
not find the value but I just know it is
a positive that means number just after
this this lobe is caused something like
this I'll take a number number just less
than 1 there is 0.9 lynch fitting there
is nothing but 3 of 0.9 square minus 1
so this becomes 3 or 0.8 1 minus 1 you
see this is less than 0 right this is
less than 0 less than 0 that means the
slope is negative something like this
that means my curve here is something
like this correct that means it is a
minima value why because the curve is in
the Uschi minima let us talk about minus
1 I know that F of minus 1 is there so
let's suppose my slope here is 0 let us
take a value that is just greater than
minus 1 that is minus 0.9 you should
take the slope of this this is nothing
but 3 or minus 0.9 square minus 1 that
is 3 or 0.8 1 minus 1 here also if you
see this less than 0 that means here the
slope is like this next is if I take a
value just left of this that is there
suppose minus 1 point 1 so this value
will come out to be 3 or 1 point 1
square minus 1 minus 1 point 1 square so
this is nothing but 3 into 1 point 2 1
minus 1 you see this guy is greater than
0 same same value actually here so here
the slope is positive this is the
position so here if you see the curve we
get is something like this
so the total curve is something like
this actually we have correct so you can
see that this guy is a maxima guy and
this guy is a minimum this is the
maximum and this guy's mean how because
here we found dad at at this point he
found that slope at this is zero slope
at this guy is positive slope at this
guy is negative so we have curve like
this in this case he found slope at this
guy is zero slope at this guy was
negative and slope at this guy's you see
positive bias low positive because if
you see this angle which x axis makes
angle 0 to 90 in this case the angle
which this guy makes with this guy is
this guy that x axis is more than 90 and
less than my nighty so that's why slope
negative slope ass and this we are able
to find the local maxima and minima
correct if you want to verify again
verify also so f of 1 will be nothing
but 1 cube minus 3 into 1 plus 3 1 Q
minus 3 into 1 plus 3 that is 1 F of
minus 1 is nothing but minus 1 Q minus
3/2 minus 1 plus 3 that is if you see
this becomes 3 plus 3 6 minus 1 that is
5 so if you see F of 1 is 5 minus 1 is 5
this guy is 5 and this guy is nothing
but but this guy is a maximum but this
guy so let us take one more example so
he and also we have to find the maxima
minima value that the continuous
function so there is no point where it
is non differentiable so I have the only
point of critical points of F dash X
equal to 0 so here also you define F
dash next equal to 0 so I have done X is
nothing but X 2 X cube becomes 2 into 3
x squared 6x square becomes 6 into 2 X 6
X becomes
six into one and five becomes insane
knowing math 6x squared minus 12x plus
six take six common this is nothing but
six into X square minus 2x plus 1 is
nothing but is six into X minus one
square this is required to zero my X is
equal to one is the only critical form
now at X is about the one I have to find
whether it is maxima or minima same
thing i can do what i can do is i have i
can find f dash of 1 point 1 and F dash
of 0.9 if you find f dash of 1 point 1
this is nothing but 6 into 1 point 1
minus 1 square so this is greater than
zero correct join this this into 0.9
minus 1 square and this is also visible
if you see the sign is not changing sign
is not changing that means at this point
of inflection there is no Maxima minima
this is point detection correct why
because if you see in both the cases the
slope is positive you take number left
of 1 or right of 1 in both the cases it
is smooth is positive so it is point of
inflection thank you
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