Question

Find the extreme values (if any) of the function f(x,y,z) = x^2 + 2y^2 subject to the constraint x^2 + y^2 -z^2 = 1.

Answer 1

Here basically we use the Langrange multiplier method.

We have function Z= f(nry) = n°+2y and the constraint +4-2=1 - z = n²+ y²1 ~ z-n?-y?+1=0. Now we use will method: the langrange Multiplier Let L= z-n² y* ++ [z?-n2-9°+1] Now doft word n In = -2n An . (2n+ dn) Ln= Now diff wort 9 Ly = -hy - dy Now dest wart zi L2= 1 +12 Now eething Ln. Ly, Iz to o.

bn zon(2+1)=0 It n70, then d=-2. then Ly= -y (6+1)=0 yao (:: 1#-4) then L2=1+177=42 'o=u t non le lx==(2+)-0. 172, then no then ly = -4 ( 4 + 1) = 0 + → |--4. .*, , = 7/1 = 2 k2p+1=27 byt Now we have +4-2=1 II 1= -2, y=0,2 = 4 then m2 +0-3-91 itaa gla 2 thus point is (+35121 0,"n)

It 47-4, M-0, Z=14, then in - n't y' - 2²=1 ► 0+ ye ) = y=1+1 16 ☺ & Jo 4 thus the point (01 + 17, 14). Now to check maximume and minimums, evaluate the function need to at pointe 5,a) and and Cavo, - Here we can lee for (ou + Sirius) we have = 0+19.2 רן 2 f(n. y) = n*+ 2y? and for (+5/2,01') (1,4)= m+ 73° = 65/4 +2.0=25/4
Thus the maximum value of f is 17/2 and the minimum value is 5/4