How do I find the max and min values of f using Lagrange multipliers.
f(x,y)=y^2 - 4x^2 subject to x^2 + 2y^2=4
f(x,y)=y^2 - 4x^2 subject to x^2 + 2y^2=4
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min/max f(x,y)=y²−4x² subject to g(x,y) = x²+2y²−4 = 0
The Lagrangian stationarity conditions are
∂f/∂x − λ∂g/∂x = 0 → −8x−2λx = 0 → x(4+λ) = 0 … (i)
∂f/∂y − λ∂g/∂y = 0 → 2y−4λy = 0 → y(1−2λ) = 0 … (ii)
x²+2y²−4 = 0 … (iii)
From (i) x=0 or λ=−4 and from (ii) y=0 or λ=1/2
Both x & y cannot be zero because of (iii)
Both x and y cannot be non-zero because then (i) and (ii) would imply different λ
If x=0 then from (iii) y=±√2 and f=2 … this is maximum
If y=0 then from (iii) x=±2 and f=−16 … this is minimum
The Lagrangian stationarity conditions are
∂f/∂x − λ∂g/∂x = 0 → −8x−2λx = 0 → x(4+λ) = 0 … (i)
∂f/∂y − λ∂g/∂y = 0 → 2y−4λy = 0 → y(1−2λ) = 0 … (ii)
x²+2y²−4 = 0 … (iii)
From (i) x=0 or λ=−4 and from (ii) y=0 or λ=1/2
Both x & y cannot be zero because of (iii)
Both x and y cannot be non-zero because then (i) and (ii) would imply different λ
If x=0 then from (iii) y=±√2 and f=2 … this is maximum
If y=0 then from (iii) x=±2 and f=−16 … this is minimum