Question

For x+y=8, the maximum value of xy is:

• A. 4
• B. 8
• C. 32
• D. 16

Answer 1

The Correct Option is D

To find the maximum value of \( xy \) given the equation \( x+y=8 \), we can use the method of expressing one variable in terms of the other and then using calculus or another mathematical tool for optimization. Here's a step-by-step breakdown:

1. Solve for one variable in terms of the other using the equation \( x+y=8 \):

\( y = 8 - x \)

2. Substitute \( y \) in the expression \( xy \):

\( xy = x(8-x) \)

3. Now, express \( xy \) as a quadratic function:

\( xy = 8x - x^2 \)

4. To find the maximum value, take the derivative of \( xy \) with respect to \( x \) and set it equal to zero:

\(\frac{d(xy)}{dx} = 8 - 2x \)

\( 8 - 2x = 0 \)

Solve for \( x \):

\( x = 4 \)

5. Substitute \( x = 4 \) back into the equation for \( y \):

\( y = 8 - 4 = 4 \)

6. Calculate \( xy \) using these values:

\( xy = 4 \times 4 = 16 \)

Therefore, the maximum value of \( xy \) is 16.