This question has ties to probability, but the question I am asking involves Calculus. See attachment.
I was able to construct A)... or at least I think so. See other attachment as I'm not quite familiar with latex.
As you found, $\displaystyle P(x) = \binom{10}{x}\pi^x(1 - \pi)^x$.
If $\displaystyle x = k$, then $\displaystyle P(x) = P(k) = \binom{10}{k}\pi^k(1 - \pi)^k$.
You are told to think of your function as a function of $\displaystyle \pi$. Therefore, let $\displaystyle P(\pi) = \binom{10}{k}\pi^k(1 - \pi)^k$.
Absolute maxima occur at critical numbers $\displaystyle c$ such that $\displaystyle f'(c) = 0$ or $\displaystyle f'(c)$ is undefined. Therefore, let's find $\displaystyle P'(\pi)$.
$\displaystyle P'(\pi) = \frac{d}{d\pi}[P(\pi)] = \frac{d}{d\pi}[\binom{10}{k}\pi^k(1 - \pi)^k] = \binom{10}{k}\frac{d}{d\pi}[\pi^k(1 - \pi)^k]$
Note $\displaystyle \binom{10}{k}$ can be 'factored' out of the derivative operator because it is a constant (scalar multiple).
$\displaystyle \binom{10}{k}\frac{d}{d\pi}[\pi^k(1 - \pi)^k] = \binom{10}{k}(\frac{d}{d\pi}[\pi^k] \times (1 - \pi)^k + \pi^k \times \frac{d}{d\pi}[(1 - \pi)^k])$
Note the use of the product rule because $\displaystyle \pi^k(1 - \pi)^k$ is a product.
$\displaystyle \binom{10}{k}(\frac{d}{d\pi}[\pi^k] \times (1 - \pi)^k + \pi^k \times \frac{d}{d\pi}[(1 - \pi)^k]) = \binom{10}{k}(k\pi^{k - 1}(1 - \pi)^k + k\pi^k(1 - \pi)^{k - 1})$
Therefore, $\displaystyle P'(\pi) = \binom{10}{k}(k\pi^{k - 1}(1 - \pi)^k + k\pi^k(1 - \pi)^{k - 1})$.
As you can see, there are no critical numbers $\displaystyle c$ such that $\displaystyle P'(c)$ is undefined. Now we only have to determine if there are any critical numbers $\displaystyle c$ such that $\displaystyle P'(c) = 0$.
$\displaystyle P'(\pi) = 0$
$\displaystyle \binom{10}{k}(k\pi^{k - 1}(1 - \pi)^k + k\pi^k(1 - \pi)^{k - 1}) = 0$
Since $\displaystyle \binom{10}{k}$ is a constant, we can eliminate it from the equation by dividing the equation by $\displaystyle \binom{10}{k}$.
$\displaystyle k\pi^{k - 1}(1 - \pi)^k + k\pi^k(1 - \pi)^{k - 1} = 0$
$\displaystyle \pi^{k - 1}(1 - \pi)^k + \pi^k(1 - \pi)^{k - 1} = 0$
$\displaystyle \pi^{k - 1}(1 - \pi)^{k - 1}[(1 - \pi) + \pi] = 0$
I factored $\displaystyle \pi^{k - 1}(1 - \pi)^{k - 1}$ from both terms on the right-hand side of the equation. Since $\displaystyle (1 - \pi) + \pi = 1$, we have:
$\displaystyle \pi^{k - 1}(1 - \pi)^{k - 1} = 0$
$\displaystyle [\pi(1 - \pi)]^{k - 1} = 0$
$\displaystyle \pi(1 - \pi) = 0$
$\displaystyle \pi = 0$ or $\displaystyle \pi = 1$
Is that not:
$\displaystyle \binom{10}{k}(\frac{d}{d\pi}[\pi^k] \times (1 - \pi)^k + \pi^k \times \frac{d}{d\pi}[(1 - \pi)^k]) = \binom{10}{k}(k\pi^{k - 1}(1 - \pi)^k +(-1)k\pi^k(1 - \pi)^{k - 1})$ which gives#
$\displaystyle P'(\pi) = \binom{10}{k}(k\pi^{k - 1}(1 - \pi)^k - k\pi^k(1 - \pi)^{k - 1})$
This gives $\displaystyle 1-2\pi=0$, or $\displaystyle \pi=\frac{1}{2}$As you can see, there are no critical numbers $\displaystyle c$ such that $\displaystyle P'(c)$ is undefined. Now we only have to determine if there are any critical numbers $\displaystyle c$ such that $\displaystyle P'(c) = 0$.
$\displaystyle P'(\pi) = 0$
$\displaystyle \binom{10}{k}(k\pi^{k - 1}(1 - \pi)^k - k\pi^k(1 - \pi)^{k - 1}) = 0$
Since $\displaystyle \binom{10}{k}$ is a constant, we can eliminate it from the equation by dividing the equation by $\displaystyle \binom{10}{k}$.
$\displaystyle k\pi^{k - 1}(1 - \pi)^k - k\pi^k(1 - \pi)^{k - 1} = 0$
$\displaystyle \pi^{k - 1}(1 - \pi)^k - \pi^k(1 - \pi)^{k - 1} = 0$
$\displaystyle \pi^{k - 1}(1 - \pi)^{k - 1}[(1 - \pi) - \pi] = 0$
I factored $\displaystyle \pi^{k - 1}(1 - \pi)^{k - 1}$ from both terms on the right-hand side of the equation.
However, you still have to take into account if $\displaystyle \pi^{k - 1}(1 - \pi)^{k - 1}$ is zero.
Hence three values for pi. These will be either maxima or minima, however, note that when pi is 0 or 1, $\displaystyle P(\pi)$ is zero, regardless of what your value of k is.$\displaystyle \pi^{k - 1}(1 - \pi)^{k - 1} = 0$
$\displaystyle [\pi(1 - \pi)]^{k - 1} = 0$
$\displaystyle \pi(1 - \pi) = 0$
$\displaystyle \pi = 0$ or $\displaystyle \pi = 1$
To find out which is max and which is min explicitly, you should get the second derivative of $\displaystyle P(\pi)$ with respect to pi, and when it is negative, the function will be maximised.
As you don't know what k which value of pi is a max/min may also vary with k.