a For fixed n are there values of p0

a) For fixed n, are there values of p(0<=p<=1) for which V(X)=0? Explain why this is so.
b) For what value of p is V(X) maximized? [Hint: Either graph V(X) as a function of p or else take a derivative.]

Solution

a)

As

V(x) = n p (1 - p)

Then if p = 0 or 1, V(x) = 0. [ANSWER, p = 0, 1]

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b)

V(x) = n p (1 - p) = n p - np^2

Taking the derivative,

V\'(x) = n - 2np = 0

--> p = 1/2

By second derivative test,

V\'\'(x) = -2n < 0, whcih makes it a relative maximum.

Thus, as at the extremes, V(x) = 0, then it is an absolute maximum.

Thus, V(x) is maximized when p =1/2 [ANSWER]