Thread: Lagrange multipliers

Q1) Find the ABSOLUTE MAXIMUM AND MINIMUM of the function f(x,y) = 4x^3 +y^2 subject to the constraint 2x^2 +y^2 =1 .

Q2) Find the MAXIMUM AND MINIMUM of the function f(x,y) = 4x^3 +y^2 subject to the constraint 2x^2 +y^2 =1 .

Are they asking to solve for the same thing? Do they have the same answer?
Please advise me how to complete them. Thank you very much.

Originally Posted by littlemu

Q1) Find the ABSOLUTE MAXIMUM AND MINIMUM of the function f(x,y) = 4x^3 +y^2 subject to the constraint 2x^2 +y^2 =1 .

We have,
f(x,y)=4x^3+y^2
c(x,y)=2x^2+y^2 with condition c(x,y)=1.

Thus,
grad (f) = k*grad (c)

Thus,
<12x^2,2y> = k * <4x,2y>

Thus,
12x^2 = 4kx
2y = 2yk
2x^2+y^2=1

Simplify them by canceling the constants,
3x^2 = kx (1)
y=yk (2)
2x^2+y^2 = 1 (3)

Look at equation (2),
It tells us:
y=0 or k=1.

y=0
Then, (3) tells us that,
2x^2 = 1
Thus, x=+/- sqrt(2)/2

k=1
Then, (1) tells us that,
3x^2 = x
Thus,
x=0 or x=1/3.

If x=0 then (3) tells us that y=+/- 1.
If x=1/3 then (3) tells us that y=+/- sqrt(7)/3

Thus, all the critical points are:
(-sqrt(2)/2,0)
(sqrt(2)/2,0)
(0,1)
(0,-1)
(1/3, -sqrt(7)/3)
(1/3), sqrt(7)/3)

Check each one for function:
f(x,y) = 4x^3 +y^2
To see the maximum and minimum.

Thank you very much