Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. (If an answer does not exist, enter DNE.) f(x, y, z) = x2y2z2; x2 + y2 + z2 = 4

  by using Lagrange multipliers

Step-by-step explanation:

Given the function     ……..(1) subject to the condition

Ф(x, y, z) =   ...…..... (2)

Step :1

Form Lagrange's function

F(x ,y, z) = f(x, y, z) + α Ф(x, y, z)

F(x ,y, z) =   ……..(3) here 'α' is the Lagrange's

multipliers

Differentiating partially with respective to 'x' we get and equating zero

on simplification , we get

    .....(a)

Differentiating partially with respective to 'y' we get and equating zero

on simplification , we get

    …. (b)

Differentiating partially with respective to 'z' we get and equating zero

on simplification , we get

  ….. (c)

Step :2

Equating (a) and (b) , we get

cancellation on both sides,we get

           ….. (d)

Equating (b) and (c) , we get

cancellation on both sides, we get

      ….   (e)

Equating (c) and (a) , we get

cancellation on both sides, we get

   …. (f)

From   (d) ,  (e), and  (f)

     ….  (4)

Using (2) and  (4) condition

and similarly  and

Therefore

The

Step 3:-

Maximum value

the maximum value = 27 /64

x=6 and x=3

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