Question

If \(x - 2y = 4\), then the minimum value of \(xy\) is

• A. \(-2\)
• B. 0
• C. 1
• D. \(-3\)

Hint:

For a quadratic expression \(ay^2 + by + c\) with \(a > 0\), minimum occurs at vertex \(y = -\frac{b}{2a}\).

Answer 1

The Correct Option is A

To find the minimum value of \(xy\) given the equation \(x - 2y = 4\), we can use concepts from algebra and calculus. We start by expressing one variable in terms of the other from the given equation.

1. Rearrange the given equation \(x - 2y = 4\) to express \(x\) in terms of \(y\):
2. \(x = 2y + 4\)
3. Substitute \(x = 2y + 4\) into the expression for \(xy\):
4. \(xy = y(2y + 4) = 2y^2 + 4y\)
5. To find the minimum value of \(2y^2 + 4y\), we take the derivative with respect to \(y\) and set it equal to zero:
6. \(\frac{d}{dy}(2y^2 + 4y) = 4y + 4\)
7. Set \(4y + 4 = 0\) to solve for \(y\):
8. \(4y + 4 = 0 \Rightarrow 4y = -4 \Rightarrow y = -1\)
9. Substitute \(y = -1\) back into the expression for \(x = 2y + 4\):
10. \(x = 2(-1) + 4 = 2\)
11. Therefore, \(xy = 2 \cdot (-1) = -2\)

The minimum value of \(xy\) is \(-2\). Hence, the correct answer is \(-2\).