How to find the absolute maximum of f(x,y)=x^2+y^2 at which 0<=x,y and 5x+9y<=9

Question

Everything is in the title, I don't really know how to start with this one :( I know the absolute minimum is (0,0) but don't know how to even begin with absolute maximum.

Restating equation just in case:

f(x,y)=x²+y² at 0<=x,y and 5x+9y<=9

Answer 1

Here, our constraint is that x and y are non-negative and that 5x + 9y <= 9. Thus, our absolute maximum will be achieved when 5x + 9y = 9.

From here, we can plug into f(x, y) and take derivative to find the maximum like such:

5x + 9y = 9

9y = 9 - 5x

y = 1 - 5x/9

f(x, y) = x^2 + y^2

= x^2 + (1 - 5x/9)^2

= x^2 + 25x^2/81 - 10x/9 + 1

=106x^2/81 - 10x/9 + 1

We see that this is a parabola with positive x^2 coefficient so the maximum value must be at the borders, which are 0 and 9/5 (due to both of the constraints). So trying both of these with the equation 5x+9y=9 we get

x= 0, y=1 so f(x, y) = 1

x=9/5, y=0 so f(x, y) = 81/25