Question

5 among all pairs of numbers whose difference is 14, find a pair whose product is as small as possible. what is the minimum product?

Explanation:

Step1: Define variables and relation

Let the two numbers be $x$ and $y$, with $x - y = 14$, so $x = y + 14$.

Step2: Set up product function

The product $P = x \cdot y = (y + 14)y = y^2 + 14y$.

Step3: Find vertex of quadratic

For $P(y)=ay^2+by+c$, vertex at $y=-\frac{b}{2a}$. Here $a=1$, $b=14$, so $y=-\frac{14}{2(1)}=-7$.

Step4: Find the other number

Substitute $y=-7$ into $x=y+14$: $x=-7+14=7$.

Step5: Calculate minimum product

$P = 7 \times (-7) = -49$.

Answer:

The pair is 7 and -7, with a minimum product of $\boldsymbol{-49}$.