C2 Differentiation

Hi there,

Got a weird question I had a few weeks ago and I can't get my head around, wondering if someone could help out a little.

Question is...:
Two real numbers x and y are such that 2x + y = 100. Find the maximum value of the product of the two numbers.

Anyone got any idea? The question itself was worth [5] marks.

Cheers.

Reply 1

latentcorpse

x= (100-y)/2 y=100-2x

xy = (50-y/2)(100-2x)=5000-100x-50y+xy

differentiate implicitly with respect to x

-100 -50dy/dx +y +xdy/dx did the last term with product rule

group together dy/dx

dy/dx(x-50) +y-100=0

so

dy/dx = (100-y)/(x-50)

Reply 2

KAISER_MOLE

latentcorpse, what on earth are you doing? Sort it out good sir.

Anyway.

100-2x = y

=> xy = 100x - 2x² = P

dP/dx = 100 - 4x

maximum occurs at turning point, at x=25

this makes y=50, and the maximum product 1250.

Reply 3

Xtrm2Matt

Cheers latent. I would've never gotten that in a million years.

It's a shame I didn't get a mark for my sarcastic answer of (2x49) + 2 = 100 :wink:

Cheers again :smile:

-edit-

Oh my (@ above post). A friend told me he got "50" as shown in your post from y = 50 which seems right to be right :smile:

Cheers Kaiser.

Reply 4

latentcorpse

yeah i didnt do the whole maximum thingy -- oopsie!

while since ive done this lol and im stupid

Reply 5

latentcorpse

my method should actually work too

Reply 6

Xtrm2Matt

Cheers once again.

Could someone have a quick shot at this Q for me. Only managed to get 2 marks on this (out of 6) o_0, was the only thing I knew how to do.

Find the turning point of the curve y = x₃ - 3x₂ + 3x - 2, and determine whether it is a local maximum, local minimum or stationary point of inflection.

I only got this far:
y = x₃ - 3x₂ + 3x - 2
dy/dx = 3x₂ - 6x +3
d₂y/dx₂ = 6x - 6

I know "something" has to be subbed into "something" but thats about it :P
As said, was out of 6 marks and but managed to get 2 for working that out ^

Cheers again.

Reply 7

Malik

dy/dx=3x²-6x+3

turning point dy/dx=0
3x²-6x+3=0
x²-2x+1=0
(x-1)(x-1)=0
x=1 I think at this point you should reconize it as a point of inflexion but you could go on...

d²y/dx²= 6x-6
when x=1
6-6=0
d²y/dx²=0 either max min or point of inflexion

d³y/dx³=6
d³y/dx³ =/= 0
Therefore point of inflexion.