i. The Lagrangian is
[tex]L(x,y,\lambda)=81x^2+y^2+\lambda(4x^2+y^2-9)[/tex]
with critical points whenever
[tex]L_x=162x+8\lambda x=0\implies2x(81+4\lambda)=0\implies x=0\text{ or }\lambda=-\dfrac{81}4[/tex]
[tex]L_y=2y+2\lambda y=0\implies2y(1+\lambda)=0\implies y=0\text{ or }\lambda=-1[/tex]
[tex]L_\lambda=4x^2+y^2-9=0[/tex]
We have [tex]f(0,-3)=9[/tex], [tex]f(0,3)=9[/tex], [tex]f\left(-\dfrac32,0\right)=\dfrac{729}4=182.25[/tex], and [tex]f\left(\dfrac32,0\right)=\dfrac{729}4[/tex], so [tex]f[/tex] has a minimum value of 9 and a maximum value of 182.25.
ii. The Lagrangian is
[tex]L(x,y,z,\lambda)=y^2-10z+\lambda(x^2+y^2+z^2-36)[/tex]
with critical points whenever
[tex]L_x=2\lambda x=0\implies x=0[/tex] (because we assume [tex]\lambda\neq0[/tex])
[tex]L_y=2y+2\lambda y=0\implies 2y(1+\lambda)=0\implies y=0\text{ or }\lambda=-1[/tex]
[tex]L_z=-10+2\lambda z=0\implies z=\dfrac5\lambda[/tex]
[tex]L_\lambda=x^2+y^2+z^2-36=0[/tex]
We have [tex]f(0,0,-6)=60[/tex], [tex]f(0,0,6)=-60[/tex], [tex]f(0,-\sqrt{11},-5)=61[/tex], and [tex]f(0,\sqrt{11},-5)=61[/tex]. So [tex]f[/tex] has a maximum value of 61 and a minimum value of -60.
