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, .
If , then .
You are told to think of your function as a function of . Therefore, let .
Absolute maxima occur at critical numbers such that or is undefined. Therefore, let's find .
Note can be 'factored' out of the derivative operator because it is a constant (scalar multiple).
Note the use of the product rule because is a product.
Therefore, .
As you can see, there are no critical numbers such that is undefined. Now we only have to determine if there are any critical numbers such that .
Since is a constant, we can eliminate it from the equation by dividing the equation by .
I factored from both terms on the right-hand side of the equation. Since , we have:
or
Is that not:
which gives#
This gives , orAs you can see, there are no critical numbers such that is undefined. Now we only have to determine if there are any critical numbers such that .
Since is a constant, we can eliminate it from the equation by dividing the equation by .
I factored from both terms on the right-hand side of the equation.
However, you still have to take into account if is zero.
Hence three values for pi. These will be either maxima or minima, however, note that when pi is 0 or 1, is zero, regardless of what your value of k is.
or
To find out which is max and which is min explicitly, you should get the second derivative of 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.