Here’s a structured explanation of the problem and the function you need to work with:

---

1. Problem Context (Abstract Understanding)

The problem involves optimizing a pivot selection strategy in sorting algorithms like quicksort. The goal is to minimize the number of comparisons $C(n, k)$, where $k$ determines the position of the pivot as a fraction of the array size $n$ (e.g., $p = k \cdot n$).

---

2. Function Definition $C(n, k)$

For quicksort, the total number of comparisons depends on two parts:

1. Comparisons to partition the array at the pivot $k \cdot n$.
2. Recursive comparisons for the sub-arrays on either side of the pivot.

The continuous approximation for $C(n, k)$ is given by:

$C(n, k) = n + k \cdot C(k \cdot n, k) + (1 - k) \cdot C((1 - k) \cdot n, k)$

For simplicity, assume that $C(x, k) = x \ln x$ (an approximation for the number of comparisons in sorting $x$ elements). Then, substituting $C(x, k) = x \ln x$ into the recursion gives:

$C(n, k) = n + k \cdot (k \cdot n \ln (k \cdot n)) + (1 - k) \cdot ((1 - k) \cdot n \ln ((1 - k) \cdot n))$

Simplify:

$C(n, k) = n + k^2 n \ln(k) + k^2 n \ln(n) + (1 - k)^2 n \ln(1 - k) + (1 - k)^2 n \ln(n)$

Combine terms:

$C(n, k) = n + n \ln(n) \cdot (k^2 + (1 - k)^2) + n \left[ k^2 \ln(k) + (1 - k)^2 \ln(1 - k) \right]$

Further simplification yields:

$C(n, k) = n (1 + \ln(n) (k^2 + (1 - k)^2) + k^2 \ln(k) + (1 - k)^2 \ln(1 - k))$

---

3. What You Need to Do

Focus on minimizing the inner term involving $k$:

$f(k) = (1 + \ln(n)) (k^2 + (1 - k)^2) + k^2 \ln(k) + (1 - k)^2 \ln(1 - k)$

---

4. Steps to Minimize $f(k)$

1. Differentiate $f(k)$ with respect to $k$ to find $f'(k)$.
2. Set $f'(k) = 0$ to find critical points.
3. Use the second derivative test $f''(k)$ to verify whether the critical points correspond to a minimum.

---

Simplified Version to Work On

You only need to minimize:

$g(k) = k^2 \ln(k) + (1 - k)^2 \ln(1 - k)$

Derivative of $g(k)$

1. Differentiate $g(k)$: $g'(k) = 2k \ln(k) + k - 2(1 - k) \ln(1 - k) - (1 - k)$
2. Solve $g'(k) = 0$ to find $k$.
3. Use the second derivative $g''(k)$ to verify the result.

---

Let me know if you need further details on any step!