SOLUTION: The positive variables x and y are such that x^4y=32. A third variable z is defined by z = x^2 + y Find the values of x and y that give z a stationary value and show that this val

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Question 1193060: The positive variables x and y are such that x^4y=32. A third variable z is defined by z = x^2 + y
Find the values of x and y that give z a stationary value and show that this value of z is a
minimum.

Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


This is fairly straightforward if you know basic calculus....

but apparently I didn't, so I deleted my response.

See the solution from the other tutor


Answer by ikleyn(52794)   (Show Source): You can put this solution on YOUR website!
.


            The solution by @greenestamps needs to be corrected.

            I came to bring a correct solution.



 --> 

 =        (1)

 =             (2)

 =             (3)


The stationary point is where the derivative is zero.

 = 

 = 

 = 

x = 2    (actually, x = +/- 2, but since we consider everything in positive numbers, we take x = 2).

At the stationary point,  and  =  =  = 


    The stationary point is a minimum if the second derivative at the point is positive; 
    or it is a maximum if that derivative is negative.  
    At x = 2, the second derivative is OBVIOULSLY positive (it is clear without any calculations)


So the stationary point is a minimum.


ANSWER: z has a stationary point that is a minimum when x = 2 and y = 2.


To make this result visually verifiable, I prepared a plot below.



    

            Plot  z =  +   (see formula (1)

Solved.

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