L22 : Second Derivative Test - Application of Derivatives, Maths, Class 12 Video Lecture

hello friends this video an application
of derivatives part 22 is brought you by
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before watching this video please make
sure that you have watched part 1 to
part 91 till now what we have seen is
finding the slope at we have to find f
dash X and then you to find f dash X
plus h there is number just left to X
and F dash X minus H this number just
write to X and compare another little
tricky little not trick it cumbersome
lot of calculations involved to make our
life simple we'll go for a second
derivative test second derivative test
will easily tell you whether it is a
local maxima minima so if F is a
function defined in the interval L again
same thing with C is any number on this
interval and F is twice differentiable
please note this condition has to be
true then only we can apply this because
it is not right why differentiable will
not be able to use second differentiable
tail so the test says that again C is a
point in FX XC is equal to 0 that means
C the critical point C can be local
maxima C can the local minima or C can
be point of inflection also correct but
we do not know to find that if you find
F Double Dash C and if this is less than
0
mine C is local maxima please note it is
less than 0 so it is maximum its other
way round actually the same case if my F
dash C is 0 I don't know it is local
maxima or minima or point of inflection
what I do is I find f SS C if my F
Double Dash C is greater than 0 then it
is a local minima this test phase if f
double dash C is 0 in this case again we
have to go back to the first derivative
long test and then we have to find
whether it is a local Maxima minima
correct they are going to find f dash of
XI plus h F dash of C minus H where H is
very less number and then you to see the
slope is changing which direction is
changing and then we can find is local
small minimal but there's no cumbersome
task so we found this second derivative
test
you just have one differentiate if it is
less than zero please note less than
zero means maximum greater than zero
anymore equal to zero that means the
test is feeling we have to go to the
first derivative test let's see how this
is actually I'll tell you graphically
how then UCF Devin - C is less than 0 it
is maxima and F Double Dash C is greater
than zero it's minimal let's see how it
is let's take this graph and let's
assume my function is something like
this right but this is the maximum so
this is my FX graph for this FX - since
I have to find f double dash X let me
draw f dash X graph also so the same
graph now if you see at this point my
slope F dash X is 0 right
so this point f dash X will be 0 these
points F dash X is negative so these
points my F this X will be something
like this negatives all negative values
the point before this list about this is
the Point C actually right so point
before C my F dash X is positive so all
these guys will have some positive L so
it will have it will graph like this
actually doing something like this this
is the graph of f dash X actually if you
plot the values you get the graph of F
dash if FX is equal to this f dash X
will be something like this now what is
slope of this slope of this is nothing
but f SS x if you see slope of this this
is negative slope correct because this
makes the angle is more than 90 and less
and already correct so what you can see
is if it affects I got f dash X and F
dash X slope is nothing but negative by
because for this graph see for this
graph the slope is negative that is f
Double Dash X is negative similarly
in this fashion if I have a minima point
like this my graph is like this FX and C
is the point of minima for the same
graph if I draw f dash X at Point C the
slope is 0 if you go in this direction
if you take any values on this direction
the slope is positive so my slope will
be something like this positive if I
take values in this direction on the
left the slope is negative correct slope
is negative this very slope is negative
so my F does X will be something like
this since my am raising so if my am
this X is like this my f double dash X
is nothing but slope of this graph F
dash X and if you see the slope of this
is also so I can say that if my f double
dash X is positive it is minimum and if
it is negative it is that is how
graphically to improve this see just
understand the concepts don't learn all
this because if you understand the
concepts you will not never forget it
same thing here if you have this graph
like this is a maximal end of graph for
this graph I have to find f double dash
X so better find F dash X first I
plotted the f dash X kind of graph now
if I see f double dash is nothing but
slope of this graph F dash X and if you
see it is negative that means if F word
double dash X is negative I am talking
about the point of maximum similar for
minima if my F dash F X graph is like
this I have to find F Double Dash eggs
before find f double dash X the first
thing is I have to find f dash X ice do
the graph of f dash X and then F Double
Dash X is nothing but slope of this and
that is positive
let's take one example on this we have
to find the local maxima and minima if
any for these two functions using my
second derivative test so second
derivative test is it says that you have
to find F Double Dash things
correct since this is a continuous
function and is open interval actually
so I am concerned only about
differentiable Part F X F dash X bar
non-differential part is not inspection
because doesn't exist so let me find for
this first F dash X and there is nothing
but 2 X and let me find F double dices
and that is nothing but if you see this
guy is positive for any number that is
greater than 0 so what I will get is
minima okay so in this f dash x is 0 I
get X ends 0 and F Double Dash X that is
f double dash 0 is nothing but 2 that is
fitter than 0 so it is minimal minimal
sorry it is minimum so 0 is a point of
name see what I've done here is my FX
equal to X square f dash X equal to X my
f dash X has to be equal to 0 for
critical points that is 2 X has to be
equal to 0 or X is equal to 0 is my
critical point now I have to find
whether X is equal to 0 is maxima or
minima or
point of inflection right three possible
is fine whether if X is going to 0 it is
maximum minimum or under inflection I
have to find f SS X or F divided by 0
actually and F divided by 0 is nothing
but 2 because F Double Dash X is 2 is
independent of 0 and this guy is greater
than 0 so that is greater than 0 I can
say that Maxima sorry it is minimal
alright one part is done similarly next
part my GX is equal to X cube minus 3 X
I have to find the points critical
points let me buy in G dash X that
becomes 3 X square minus 3 and I will
make it as 0 because I define critical
points so this means I get 3
x squared X plus 1 into X minus 1 is
equal to 0 or X is equal to 1 n minus 1
are my critical points now I have to
find whether at these points the
function is Maxima or minima so I take 1
first write x is equal to 1 X equal to 1
you take let me find G dash Double Dash
expressed G divide s is nothing but 3
into X square becomes 2 x3 become 0 is 6
X so G double dash of 1 will be 6 into 1
that is 6 and that is greater than 0
since it is greater than 0 it is a point
of minimum similarly when X is equal to
minus 1 G double dash minus 1 will be
nothing but 6 into minus 1 because 6 X
and that is nothing but minus 6 that is
less than 0 since it is less than 0 X is
equal to minus 1 is the point of maximum
correct so is this question solve down
no why because the question says find
the local maxima and minima they are not
the question is not asking me to find
the point of local maxima minima the
question is asking me to find the local
maxima minima so let us pour this to
this my point of local Maxima is is
minimized 0 here actually
so my local minima will be F of 0 that
is 0 square that is 0 so my local
minimizer in this case my point of local
minima point of local minima is one else
a point of minimum is 1 so minimum value
is how much G of 1 is nothing but 1 Q
minus 3 into 1 and that is minus
similarly point of maximum is equal to
minus 1 correct
so my maximum value will be nothing but
G of minus 1 and that is nothing but X Q
minus 3x that is minor
one hue minus 3 into minus 1 and that is
nothing but minus 1 plus 3 and that is
nothing but 2 so if you see point of
minimize minus 2 endpoint of maximize
and that is my but correct so what we
have done in this case first we found
the critical points you see here where
it can be Maxima or minima but now we
are not sure where this guy is a Maxima
or minima so we found G double dash X
and we found this this guy is Maxima at
this guy's minimum when on whether G
double dash is greater than 0 or less
than 0 it is greater than 0 it is minima
less than 0 Maxima and then since the
question asked me to find local maximum
minimum i form the local maxima and
local minima thank you
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