Question

Find the greatest solution for x+y when x^2+y^2 = 7, x^3+y^3=10

Answer 1

4

Explanation:

set

`f(x,y)=x+y\\`

constrain:

`g(x,y)=x^2+y^2 = 7\\h(x,y)=x^3+y^3=10`

Partial derivatives:

`f_(x)=1\\f_(y) =1 \\g_(x)=2x \\g_(y)=2y\\h_(x)=3x^2 \\h_(y)=3y^2`

Lagrange multiplier:

`grad(f)=a*grad(g)+b*grad(h)\\`

`\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]`

4 equations:

`1=2ax+3bx^2\\1=2ay+3by^2\\x^2+y^2=7\\x^3+y^3=10`

By solving:

`a=4/9\\b=-2/27\\x+y=4`

Second mathod:

Solve for x^2+y^2 = 7, x^3+y^3=10 first:

`x=(1)/(2) -(√(13))/(2) \ or \ y=(1)/(2) +(√(13))/(2) \\x=(1)/(2) +(√(13))/(2) \ or \ y=(1)/(2) -(√(13))/(2) \\x+y=-5\ or\ 1 \or\ 4`

The maximum is 4