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# differentiation questions watch

1. The 2nd one.
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2. heh thanks alot fermat, i can understand your 2nd one perfectly, but not your first, can u explain it a bit more if it's not too much trouble ?

btw is there any 'special' ways of identifying the equations in these type of questions ?
3. (Original post by ronnie)
heh thanks alot fermat, i can understand your 2nd one perfectly, but not your first, can u explain it a bit more if it's not too much trouble ?

btw is there any 'special' ways of identifying the equations in these type of questions ?
Ignore the 1st one - I'll re-do it.
4. i will try my hand at the first one

take the top point (ie the top point where the rectangle touches the xy=10 graph). this point has coordinates (x,y)
due to it being symmetrical in x and y axis, one side is 2y, and the other is 2x

the perimeter of the rectangle is therefore 4x + 4y.
and the area is (2x)(2y)

take xy=10. y = 10/x
perimeter = 4x + 40x-1
dy/dx = 4 - 40x-2
dy/dx = 0 (for min/max)
4 - 40x-2=0
x²=10
x=√10
y=√10

d²y/dx² = 80x-3
80/(√10)³>0, so it is a minimum.
MIN perimeter = 4√10 + 4√10 = 8√10

Area = (2x)(2y)
= (2x)(20/x)= 40
EDIT: Area = 40 = constant, so not min or max
5. Q1)
Look at the rectangle
The x- and y-axes are axes of symmetry. therefore P and Q are directly opposite each other on the opposite sides of the origin.

:. x1 = -x0
& y1 = -y0

Perimeter of rectangle is S = 2(x0 + |x1|) + 2(y0 + |y1|)
(this is just adding the coordinates of P and Q to get the length and height)

but |x1 | = x0
and |y1 | = y0

:. S = 2(2x0) + 2(2y0)
S = 4x0 + 4y0

since xy = 10 then
y0 = 10/x0

S = 4x0 + 40/x0
dS/dx0 = 4 - 40/x02 = 0
:. x02 = 10
x0 = √10, y0 = √10
==============

S = 4x0 + 4y0
S = 4√10 + 4√10
S = 8√10
=======

d²S/dx02 = 80/x03, which is +ve for x0 +ve, => S is a min
================================ ==============

Area
====
A = (x0 +|x1|)(y0 + |y1|)
A = 2x0.2y0
A = 4x0y0
A = 40 (xy = 10)
=====

i.e. there is no max or min - it is always a constant
================================ =======
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6. (Original post by Fermat)
i.e. there is no max or min - it is always a constant
Ahhh i see. i first did my differentiation of Area wrong (see above) and then did again and got 0. so i agree

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