Question

What is the maximum value of the function f(x, y) = 4x² +10y² on the curve x² + y² = 4 ? (a) 28 (b) 34 (c) 40 (d) 56 (e) 60

Answer 1

C) 40

Explanation:

Use Lagrange Multipliers

`\displaystyle f(x,y)=4x^2+10y^2\\g(x,y)=x^2+y^2-4\\L(x,y,\lambda)=f(x,y)-\lambda g(x,y)=4x^2+10y^2-\lambda(x^2+y^2-4)\\\\(\partial L)/(\partial x)=8x-2x\lambda\rightarrow8x-2x\lambda=0\rightarrow \lambda=4,\,\text{or}\,\,x=0\\\\(\partial L)/(\partial y)=20y-2y\lambda\rightarrow20y-2y\lambda=0\rightarrow \lambda=10,\,\text{or}\,\,y=0`

If x=0

`g(0,y)=0^2+y^2-4\\0=y^2-4\\4=y^2\\y=\pm 2`

`f(0,\pm2)=4(0)^2+10(\pm2)^2=40`

If y=0

`g(x,0)=x^2+0^2-4\\0=x^2-4\\4=x^2\\x=\pm 2`

`f(\pm 2,0)=4(\pm 2)^2+10(0)^2=16`

Therefore, 40 is our maximum at (0,2) and (0,-2)