The given problem involves determining the local maximum of the function:
$f(x) = |x|^m |x - 1|^n, \quad x \in \mathbb{R}, \, m, n \in \mathbb{N}.$

To find the local maximum value, we proceed as follows:

Step 1: Analyze the function

The function $f(x)$ is the product of two absolute values, raised to the powers $m$ and $n$:

1. $|x|^m$: This term is zero at $x = 0$, increases as $|x|$ increases, and is symmetric about $x = 0$.
2. $|x - 1|^n$: This term is zero at $x = 1$, increases as $|x - 1|$ moves away from $1$, and is asymmetric.

The combined behavior creates a local maximum at some point between $0$ and $1$, where the product achieves a balance between the two terms.

Step 2: Rewrite $f(x)$

Let $x$ be in $[0, 1]$. The absolute values simplify to: $f(x) = x^m (1 - x)^n, \quad 0 \leq x \leq 1.$

Step 3: Find the critical points

To find the critical points, compute the derivative $f'(x)$: $f'(x) = \frac{d}{dx} \left( x^m (1 - x)^n \right).$ Using the product rule: $f'(x) = m x^{m-1} (1 - x)^n - n x^m (1 - x)^{n-1}.$

Set $f'(x) = 0$: $m x^{m-1} (1 - x)^n = n x^m (1 - x)^{n-1}.$

Simplify by dividing through by $x^{m-1} (1 - x)^{n-1}$ (valid since $x \in (0, 1)$): $m (1 - x) = n x.$

Rearrange to solve for $x$: $x = \frac{m}{m + n}.$

Step 4: Compute $f(x)$ at $x = \frac{m}{m+n}$

Substitute $x = \frac{m}{m+n}$ into $f(x) = x^m (1 - x)^n$: $f \left( \frac{m}{m+n} \right) = \left( \frac{m}{m+n} \right)^m \left( 1 - \frac{m}{m+n} \right)^n.$

Simplify $1 - \frac{m}{m+n}$: $1 - \frac{m}{m+n} = \frac{n}{m+n}.$

Thus: $f \left( \frac{m}{m+n} \right) = \left( \frac{m}{m+n} \right)^m \left( \frac{n}{m+n} \right)^n.$

Step 5: Final Expression for the Local Maximum

The local maximum value is: $f_{\text{max}} = \frac{m^m n^n}{(m+n)^{m+n}}.$

Step 6: Correct Answer

From the given options, the correct answer is:
(d) $\frac{(mn)^{mn}}{(m+n)^{m+n}}$.

Let me know if you'd like further clarification or detailed steps!

Related Questions:

1. What is the derivative of $x^a (1-x)^b$ for general $a, b > 0$?
2. How do you find the critical points of a product of functions?
3. How does the symmetry of $|x|$ and $|x - c|$ affect the behavior of functions?
4. Can you derive the second derivative of $f(x)$ to confirm the local maximum at $x = \frac{m}{m+n}$?
5. How does the value of $m$ and $n$ affect the location and value of the maximum?

Tip:

When dealing with products of terms involving absolute values, carefully simplify their behavior in different regions of $x$ and use symmetry to reduce the complexity of calculations.