# Differentiation

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#1
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1 year ago
#2
(Original post by Hannahkisley)
Well the product is clearly "xy".

If you could write y in terms of x, you could make that product just a function of x, then differentiate to find the stationary point, and then find the value of the function.
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#3
(Original post by ghostwalker)
Well the product is clearly "xy".

If you could write y in terms of x, you could make that product just a function of x, then differentiate to find the stationary point, and then find the value of the function.
So I got 25x50 = 1250 is this correct?
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1 year ago
#4
(Original post by Hannahkisley)
So I got 25x50 = 1250 is this correct?
Agreed.
0
1 year ago
#5
(Original post by Hannahkisley)
So I got 25x50 = 1250 is this correct?
- You're given 2x+y=100.
- Rearrange to make y the subject: y = 100 - 2x (1)

- Multiply everything by x to get the product required "xy" : yx = 100x - 2x^2

- Let z = xy, then find dz/dx: dz/dx = 100 - 4x
- Max value of z is when dz/dx = 0: Hence 100 - 4x = 0
- Therefore x = 25 at max value.

- Sub x = 25 into (1): y = 100 - 2.(25)
- Therefore y = 50
- Hence max value of z = 25*50
- Complete to obtain correct answer.

---------------------------------------------------------------
Alternatively,
- Complete the square of z = 100x - 2x^2
- z = -2[x^2 - 50x]
- z = -2[(x-25)^2 - 25^2]
- z = 1250 - 2(x-25)^2

- Hence max value of z is when x=25, since squares cannot be negative, and so cannot minus a negative number from 1250 to make it larger.
- When x=25, z=1250.
- Complete derivation and write down logical argument to obtain correct answer.

(Apparently I'm not allowed to make sure you've followed every step logically even though you've already posted that you have the correct answer, so I've had to leave the final step to you to complete.)
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