Question

$x$ and $y$ are real numbers satisfying the conditions $2<x<3$ and $-8<y<-7$. Which of the following expressions will have the least value?

• A. $x^2 y$
• B. $x y^2$
• C. $5xy$
• D. None of these

Hint:

When $y$ is negative, maximizing the positive factor with $y$ makes the product more negative.

Answer 1

The Correct Option is A

Since $y$ is negative, expressions with larger positive $x$ multiplier will give more negative values.
$x^2 y$: $x^2$ ranges from $4$ to $9$. Multiplying by $y \approx -8$ gives range $\approx [-72, -32]$.
$xy^2$: $y^2$ positive large $\approx 64$, $x \approx 2$ to $3$, so $xy^2$ positive large — not least.
$5xy$: $5 \times$ negative product $\approx -80$ to $-70$, which is less negative than $x^2 y$ for max $x^2$. Thus $x^2 y$ is smallest.