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

- "runner b is moving at a slower rate."

- "the graphs shown are graphs of distance(km) against time(min).

the slopes of this graphs are are given by:

slope = (change in y)/(change in x)

= (change in distance)/(change in time)

this is the formula for finding the speed. so, the slope of these graphs represent the speed of the runner.

taking each division in the graph to represent 1 on both axis:

runner a ⇒ speed = (2-0)/(1-0)

= 2/1

= 2 km/min

runner b ⇒ speed = (3-0)/(3-0)

= 3/3

= 1 km/min

this shows that   runner b is moving at a slower rate."

3 qrs + 3/2 xy hope this : /

step-by-step explanation: