Chapter 13 - Section 13.8 - Lagrange Multipliers - Exercises - Page 746: 3

$39$

As we are given that $f(x,y) =49-x^2-y^2$ and the equation of constraint $g(x,y)=x+3y-10=0$ since, we have the gradient equation $\nabla =\lambda \nabla g$ Now, $ -2x=\lambda \implies x=-\dfrac{ \lambda}{2}$ and $ -2y=3 \lambda \implies y=-\dfrac{3\lambda}{2}$ Now, $g(x,y)=x+3y-10=0 \implies \lambda=-2$ and $x=1 ; y=3$ Thus, $f(x,y) =49-1-9=39$