Find the maximum and minimum values of f(x,y)=xy on the ellipse 7x^2+y^2=5.?

We want to find the maximum and minimum values of f(x,y) = xy, subject to the constraint
g(x,y) = 7x^2 + y^2 = 5.

By Lagrange Multipliers, ∇f = λ∇g ==> <y, x> = λ<14x, 2y>.

Equating like entries:
y = 14λx
x = 2λy.

Substitute the first equation into the second equation:
x = 28λ^2 x ==> x(28λ^2 - 1) = 0.

(i) If x = 0, then substituting this into g yields y = ±√5.

(ii) If 28λ^2 - 1 = 0 <==> λ = ±1/√28, then y = ±x√7.
Substitute this into g: 7x^2 + 7x^2 = 5 ==> x = ±√(5/14) and thus y = ±√(5/2).
(Note that this yields four points, one for each sign combination.)

Testing the points for extrema:
f(0, ±√5) = 0
f(±√(5/14), ±√(5/2)) = 5/√28 <---Maximum
f(±√(5/14), ∓√(5/2)) = -5/√28 <---Minimum

I hope this helps!

Parameterize ellipse as x=√(5/7).cos(t), y=√5.sin(t)

xy = (5/√7).cos(t)sin(t) = ½(5/√7).sin(2t)

Because sin(2t) varies between ±1, max xy = ½(5/√7) & min xy = −½(5/√7)