Question

Find the absolute extreme values of
f(x, y) = x^2 +x+y^2 −2y
defined on { x^2 + y^2  14 years ago

Answer 1

not the best way of doing it

Answer 2

i think u left something out

Answer 3

you set grad f =0 for inside the disc

Answer 4

grad f = ( 2x +1 , 2y-2) =zero vector

so (-1/2 , 1 ) is a candidate for max /min

Answer 5

then let g = x^2 +y^2 (the boundary )

by lagrange eqn

( 2x +1 , 2y-2 ) = A ( 2x, 2y )


so 2x +1 = 2Ax -- eqn 1

2y-2 = 2Ay --- eqn 2

now solve both eqns for A, and equate

(2x+1) / (2x) = (2y-2)/ (2y)


(2x+1)/(2x) = (y-1)/ y
y(2x+1) = 2x(y-1)

so

y=-2x

sub this into the boundary of the constraint

x^2 +4x^2 = 1
x= +- sqrt(1/5)

Answer 6

so then you have two more points

( sqrt(1/5) , - 2 sqrt(1/5) ) and ( -sqrt(1/5) , 2sqrt(1/5) )

Answer 7

so you have three points , sub them into the function to find the respective function values , then chose the largest and smallest values

Answer 8

note: the point (-1/2, 1 ) is outside the unit disc , so discard that .

Answer 9

Yeah, elec is right - I tried to do this problem an easier way but skipped a step which gives min/max points that aren't extrema of x and y.

Answer 10

oohh thankyou elecengineer.....that helpd alot
and thankyouu blacksteel :D