Question

Write an odd natural number as a sum of two integers m_1 and m_2 in a way that m_1 m_2 is maximum.

Answer 1

Answer on Question #74604 – Math – Algebra

Question

Write an odd natural number as a sum of two integers $m_1$ and $m_2$ in a way that $m_1m_2$ is maximum.

Solution

Consider the natural odd number $2m + 1, m$ is given.

Let $m_1 + m_2 = 2m + 1, m_1 \in \{0, 1, \ldots, 2m + 1\}, m_2 \in \{0, 1, \ldots, 2m + 1\}$.

Then $m_2 = 2m + 1 - m_1$.

Consider the product $m_1m_2$ as the function $f(m_1)$

$$f(m_1) = m_1(2m + 1 - m_1)$$

Take the first derivative with respect to $m_1$

$$f'(m_1) = \left(m_1(2m + 1 - m_1)\right)' = 2m + 1 - m_1 + m_1(0 + 0 - 1) = 2m + 1 - 2m_1$$

Find the critical point(s)

$$f'(m_1) = 0 \Rightarrow 2m + 1 - 2m_1 = 0 \Rightarrow m_1 = m + \frac{1}{2}$$

If $0 \leq m_1 < m + \frac{1}{2}, f'(m_1) > 0, f(m_1)$ increases

If $m + \frac{1}{2} < m_1 \leq m, f'(m_1) < 0, f(m_1)$ decreases

The function $f(m_1)$ has maximum at $m_1 = m + 1/2$.

Since $m_1$ is integer number, then we consider

$m_1 = m + 0$ or $m_1 = m + 1$

Therefore, we have that

$$2m + 1 = (m) + (m + 1)$$

If $M$ is the odd natural number, then

$$M = \frac{1}{2}(M - 1) + \frac{1}{2}(M + 1)$$

The maximum of product

$$\frac{1}{2}(M - 1)\left(\frac{1}{2}\right)(M + 1) = \frac{M^2 - 1}{4}$$

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