Point of Inflection - Differentiation, Business Mathematics & Statistics Video Lecture | Business Mathematics and Statistics - B Com

let G of X equal 1/4 X to the fourth
minus 4x to the third power plus 24x
squared for what values of X does the
graph of G have an inflection point
we'll have a point of inflection so
let's just remind ourselves what a point
of inflection is a point of inflection
is where we go from being cut where we
change our concavity or you could say
where our second derivative G prime of X
switches signs switches switches signs
so let's study our second derivative in
order to study our second derivative
let's find it so we know that G of X is
equal to 1/4 X to the fourth minus 4 X
to the third power plus 24x squared so
given that let's now find G prime of X G
prime of X is going to be equal to I'm
just going to apply the power rule
multiple times 4 times 1/4 is just 1 I'm
not going to write the 1 down it's going
to be 1 times X to the 4 minus 1 power
so 4 to the third power minus 3 times 4
is 12 X to the 3 minus 1 power X to the
second power plus 2 times 24 48 X to the
2 minus 1 or X to the first power I
could just write that as X so there you
have it I have our first derivative now
we want to find our second derivative so
G prime prime of X is just the
derivative of the first derivative with
respect to X and so more of the power
rule 3x squared minus 24x to the first
or just 24x plus 48 so let's think about
where this switches signs and this is
this is a continuous function it's going
to be defined for all X's so the only
potential candidates of where it could
switch signs are when this thing equals
zero so let's see where it equals zero
so let's set it equal to zero 3x squared
minus 24x plus 48 is equal to zero let's
see everything is divisible by 3 so
let's divide everything by
so you get x squared minus 8x plus 16
plus 16 is equal to zero and let's see
can i factor this yeah this would be X
minus 4 times X minus 4 or you could
just view this as X minus 4 squared is
equal to 0 or X minus 4 is equal to 0 so
so or where x equals 4 so G prime prime
of 4 is equal to 0 so let's see what's
happening at on either side of that
let's see if G if we're actually if
we're actually switching signs or not so
let me draw a number line here and so
this is so this is 2 3 4 5 and I could
keep going and so we know that something
interesting is happening right over here
G prime prime of 4 is equal to 0 G prime
prime of 4 is equal to 0 so let's think
about what's what the second derivative
is when we are less than 4 and so
actually let me just try G prime prime
of 0 since that'll be easy to evaluate G
prime prime of 0 well it's just going to
be equal to 48 so when we are less than
4 our second derivative G prime the
second derivative is greater than 0 so
we're actually going to be concave
upwards over this interval to the left
of 4 now let's think about to the right
of 4 2 this is a different color so what
about to the right of 4 and so let me
just evaluate what would be an easy
thing to evaluate well I could evaluate
G prime of well when I do Jeep or the
second derivative is a of 10 so G I'll
do it right over here let me do it well
I'm running a little bit of space so
I'll just scroll down so G prime prime
of 10 is going to be equal to 3 times 10
squared so it's 300 minus 24 times 10 so
minus 240 plus 48 so let's see this is
60
this is so 300 - 240 is 60 plus 48 so
this is equal to 108 so it's still
positive so on either side of for G
prime prime of X is greater than zero so
even this even though the second
derivative at x equals four is equal to
zero on either side we are concave
upwards on either side the second
derivative is positive and so and that
that was the only potential candidate so
there are no X values of X for which G
has a point of inflection x equals four
would have been a value of x at which G
had a point of inflection if we switch
if the second derivative switch signs
here if it went from positive to
negative or negative to positive but
it's just staying from positive to
positive so the second derivative is
positive it just touches zero right here
and then it goes positive again so going
back to the question for what X values
is the graph of G have a point of
inflection no X values I'll put an
exclamation mark there just for drama