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Problem 1

Solve the optimization problems. Maximize \(P=x y\) with \(x+y=10\).

Short Answer

Expert verified
The maximum value of the function \(P(x, y) = xy\) with the constraint \(x + y = 10\) is 25, which occurs when \(x = 5\) and \(y = 5\).
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Step 1: Solve the constraint for one of the variables

Let's solve the constraint \(x + y = 10\) for y: \[y = 10 - x\]

Step 2: Substitute the expression from Step 1 into the objective function

Now, substituting the expression for y from Step 1 into the objective function \(P(x, y) = xy\), we obtain \(P(x) = x(10 - x)\): \[P(x) = 10x - x^2\]

Step 3: Find the critical points of the function

To find the critical points of P(x), we need to take the first derivative and set it equal to zero: \[\frac{dP}{dx} = \frac{d}{dx}(10x - x^2) = 10 - 2x\] Now, set the derivative equal to zero and solve for x: \[10 - 2x = 0\] \[x = 5\]

Step 4: Determine the value of y

Now that we have found the value of x that maximizes P(x), we can find the corresponding value of y from the expression derived in Step 1: \[y = 10 - x = 10 - 5 = 5\]

Step 5: Determine the maximum value of P(x, y)

Finally, substitute the values of x and y found in the previous steps into the objective function to find the maximum value of P(x, y): \[P(5, 5) = 5 \cdot 5 = 25\] The maximum value of P(x, y) is 25 when x = 5 and y = 5.

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