∇f = λ∇g
∇(x² + y²) = λ∇(5x^2 + 6xy + 5y^2)
<2x, 2y> = λ<10x+6y, 6x+10y>
= λ<5x+3y, 3x+5y>
x = λ(5x+3y)
y = λ(3x+5y)
5x² + 6xy + 5y² = 8
x = 5λx+3λy
y = 3λx+5λy
xy = 5λxy + 3λy²
xy = 3λx² + 5λxy
5λxy + 3λy² = 3λx² + 5λxy
λy² = λx²
λ(y² − x²) = 0
λ(x + y)(x − y) = 0
Then either
λ = 0
Or
x = y
Or
x = −y
If λ = 0 then x=y=0
If x = y then x²=y² and
5x² + 6xy + 5y² = 8
5x² + 6x² + 5x² = 8
16x² = 8
x² = 1/2
x = ±√1/2
And y = ±√1/2
(√1/2, √1/2) or (−√1/2, −√1/2)
If x = −y then x² = y² and
5x² + 6xy + 5y² = 8
5x² − 6x² + 5x² = 8
4x² = 8
x² = 2
x = ±√2
(√2, −√2), (−√2, √2)
Test each of these answers
(0,0), (√1/2, √1/2), (−√1/2, −√1/2), (√2, −√2), (−√2, √2)
At (0,0) we have a minimum of z=0
At (√2, −√2) and (−√2, √2) we have a maximum of z=4
At each of the other two, we have an intermediate value, as these were superfluous solutions.
∇(x² + y²) = λ∇(5x^2 + 6xy + 5y^2)
<2x, 2y> = λ<10x+6y, 6x+10y>
x = λ(5x+3y)
y = λ(3x+5y)
5x² + 6xy + 5y² = 8
x = 5λx+3λy
y = 3λx+5λy
xy = 5λxy + 3λy²
xy = 3λx² + 5λxy
5λxy + 3λy² = 3λx² + 5λxy
λy² = λx²
λ(y² − x²) = 0
λ(x + y)(x − y) = 0
Then either
λ = 0
Or
x = y
Or
x = −y
If λ = 0 then x=y=0
If x = y then x²=y² and
5x² + 6xy + 5y² = 8
5x² + 6x² + 5x² = 8
16x² = 8
x² = 1/2
x = ±√1/2
And y = ±√1/2
(√1/2, √1/2) or (−√1/2, −√1/2)
If x = −y then x² = y² and
5x² + 6xy + 5y² = 8
5x² − 6x² + 5x² = 8
4x² = 8
x² = 2
x = ±√2
(√2, −√2), (−√2, √2)
Test each of these answers
(0,0), (√1/2, √1/2), (−√1/2, −√1/2), (√2, −√2), (−√2, √2)
At (0,0) we have a minimum of z=0
At (√2, −√2) and (−√2, √2) we have a maximum of z=4
At each of the other two, we have an intermediate value, as these were superfluous solutions.