Lagrange's multiplier method find the highest and the lowest point

Question

Plane $x+y+z=12$ intersects the paraboloid $z=x^2+y^2$ find the highest and the lowest point of this cross-section.

What should i do here? I need help solely when it comes to transforming this question into 'find max, min of the function on a given set'.

Answer 1

you must solve the system $$2x+\lambda(1+2x)=0$$ $$2y+\lambda(1+2y)=0$$ $$x+y+x^2+y^2=12$$ easy you will get $$x=y=-3,\lambda=-\frac{6}{5}$$ or $$x=y=2,\lambda=-\frac{4}{5}$$

Answer 2

Hint

Highest (lowest) point is when $z$ is maximal (minimal). Since you are on the paraboloid, $z$ is fixed as a function of $x,y$. Plug this back into your constraint to get to optimize $f(x,y)$ subject to $g(x,y) = 12$.