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Problem 1

# Solve the optimization problems. Maximize $$P=x y$$ with $$x+y=10$$.

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 $$y = 5$$.
See the step by step solution

## 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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