global max/min on disc

1. The problem statement, all variables and given/known data



find the global extrema on the disc [tex] x^2 + y^2 \le 1[/tex]



given the function [tex]f(x,y)=xy+5y[/tex]









3. The attempt at a solution



For the interior of the disc



[tex]\nabla f = <y,x+5>[/tex]



the critical point is (0,-5)



for the boundary of the disc

using lagrange multipliers



[tex] \left\{\begin{array}{cc}y=\lambda 2x \\ x+5 = \lambda 2y \\ x^2+y^2 =1 \end{array}\right.[/tex]



solving for lambda



[tex] \lambda = \frac{y}{2x}; \lambda = \frac{x+5}{2y} [/tex]

[tex]\frac{y}{2x}=\frac{x+5}{2y} \Rightarrow y = \pm \sqrt{x(x+5)}[/tex]

now,

[tex]x^2 + (\pm \sqrt{x(x+5)})^2 = 1 \Rightarrow x = \frac{1}{4}(\pm\sqrt{33}-5)[/tex]

subbing the x value into

[tex]y = \pm \sqrt{x(x+5)} \Rightarrow \sqrt{\frac{1}{8}(5 \sqrt{33} - 21)}[/tex]



I know to test those critical points in the original function but before I go further I want to make sure I have done everything up to it correctly