Answer:
4
Step-by-step explanation:
set
[tex]f(x,y)=x+y\\[/tex]
constrain:
[tex]g(x,y)=x^2+y^2 = 7\\h(x,y)=x^3+y^3=10[/tex]
Partial derivatives:
[tex]f_{x}=1\\f_{y} =1 \\g_{x}=2x \\g_{y}=2y\\h_{x}=3x^2 \\h_{y}=3y^2[/tex]
Lagrange multiplier:
[tex]grad(f)=a*grad(g)+b*grad(h)\\[/tex]
[tex]\left[\begin{array}{ccc}1\\1\end{array}\right]=a\left[\begin{array}{ccc}2x\\2y\end{array}\right]+b\left[\begin{array}{ccc}3x^2\\3y^2\end{array}\right][/tex]
4 equations:
[tex]1=2ax+3bx^2\\1=2ay+3by^2\\x^2+y^2=7\\x^3+y^3=10[/tex]
By solving:
[tex]a=4/9\\b=-2/27\\x+y=4[/tex]
Second mathod:
Solve for x^2+y^2 = 7, x^3+y^3=10 first:
[tex]x=\frac{1}{2} -\frac{\sqrt{13}}{2} \ or \ y=\frac{1}{2} +\frac{\sqrt{13}}{2} \\x=\frac{1}{2} +\frac{\sqrt{13}}{2} \ or \ y=\frac{1}{2} -\frac{\sqrt{13}}{2} \\x+y=-5\ or\ 1 \or\ 4[/tex]
The maximum is 4
