What we're trying to maximize is the distance from a point (x,y) to the point (0, -1) with the constraint x2 + 16y2 = 16.
So - using the distance formula for two points - our f(x, y) is √((x-0)2 + (y-(-1))2) (what we're maximizing)
And g(x,y) = x2 + 16y2 (our constraint)
Maximizing the distance between two points is equivalent to maximizing the distance SQUARED between two points. Therefore, for the sake of simplicity, f(x, y) = (x-0)2 + (y+1)2 = x2 + (y+1)2
By Lagrange, we set up the equations ∇ f(x, y) = λ ∇ g(x, y).
First, ∇ f(x, y) = < 2x, 2(y+1) > and ∇ g(x, y) = < 2x, 32y >
Equating components, we have 2x = λ2x and 2(y+1)=λ32y
The first equation tells us λ=1, which we plug into the second to get 2y + 2 = 32y
So we get y = 1/15 ≈ .0667
We plug this y-value into the constraint equation (where the point (x,y) is on!) and solve for x:
x = ± 3.991
Therefore, the points on the ellipse which are furthest away from (0, -1) are:
(3.991, 0.0667) and (-3.991, 0.0667)