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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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):
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)
