Question

Use Lagrange multipliers to minimize the function subject to the following two constraints. Assume that x, y, and z are nonnegative. Question 18 options: a) 192 b) 384 c) 576 d) 128 e) 64

Answer 1

Complete Question

The complete question is shown on the first uploaded image

Answer:

Option C is the correct option

Step-by-step explanation:

From the question we are told that

   The equation is  $f (x, y , z ) = x^2 +y^2 + z^2$

    The constraint is  $P(x, y , z) = x + y + z - 24 = 0$

Now using Lagrange multipliers  we have that  

   $\lambda = \frac{ \delta f }{ \delta x } = 2 x$  

   $\lambda = \frac{ \delta f }{ \delta y } = y$  

   $\lambda = \frac{ \delta f }{ \delta z } = 2 z$

=>       $x = \frac{ \lambda }{2}$

          $y = \frac{ \lambda }{2}$

         $z = \frac{ \lambda }{2}$

From the constraint  we have

      $\frac{\lambda }{2} + \frac{\lambda }{2} + \frac{\lambda }{2} = 24$

=>   $\frac{3 \lambda }{2} = 24$

=>   $\lambda = 16$

substituting for x, y, z

=>   x =  8

=>  y =  8

=>   z =  8        

Hence

    $f (8, 8 , 8 ) = 8^2 +8^2 + 8^2$

    $f (8, 8 , 8 ) = 192$