Find the maximum value of the function f(x,y)=ln(xy^2) subject to 2 x^2+7 y^2=8 for x>0 and y>0.
Maximum value:
Maximum value:
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∇ f = λ ∇ g ...... by the method of Lagrange Multiplers
1/x = λ 4x ...... ∂/∂x ....... eq. (1)
2/y = λ 14y .... ∂/∂y ....... eq. (2)
⇒ 7y² = 4x² ...... divide eq. (1) by eq. (2)
6x² = 8 .... plug into 2x² + 7y² = 8
x = 2/√3 , y = 4/√21
Answer: x = 2/√3 , y = 4/√21
1/x = λ 4x ...... ∂/∂x ....... eq. (1)
2/y = λ 14y .... ∂/∂y ....... eq. (2)
⇒ 7y² = 4x² ...... divide eq. (1) by eq. (2)
6x² = 8 .... plug into 2x² + 7y² = 8
x = 2/√3 , y = 4/√21
Answer: x = 2/√3 , y = 4/√21