Question

This extreme value problem has a solution with both o maximum value and a minimum value. Use Lagrange muttiplers to find the extreme values of the function subject to the given constraint.

f(x,y,z)=xy 2 ;x 2 +y 2 +z 2 =16
maximum value X =
minimum value x=

Answer 1

To find the extreme values of the function f(x,y,z)=xy^2 subject to the constraint x^2+y^2+z^2=16, we can use Lagrange multipliers.

Step-by-step explanation:

To find the extreme values of the function f(x,y,z)=xy^2 subject to the constraint x^2+y^2+z^2=16, we can use Lagrange multipliers.

1. First, let's define the Lagrange function: L(x,y,z,λ) = xy^2 + λ(x^2 + y^2 + z^2 - 16).
2. Next, we need to find the partial derivatives of the Lagrange function with respect to x, y, z, and λ. Set each derivative equal to zero and solve the resulting system of equations to find the critical points.
3. Finally, evaluate the function f(x,y,z) at each critical point to determine the extreme values.