9) Given function is :
Putting n = 250 we get,
i.e.,
i.e.,
i.e.,
Therefore, the minimum cost is $2500.
11) Let
Taking loge on both sides we get,
i.e.,
i.e.,
Then,
i.e., [Using L'hospital rule]
i.e.,
i.e., [Using L'hospital rule]
i.e.,
i.e.,
i.e.,
i.e.,
i.e.,
Therefore, limiting value of the given function is 1.
12) Given function is :
For minimum value of f(x) we must have .
i.e.,
i.e.,
i.e.,
i.e.,
Therefore, required value of x is 1.
Putting x = 1 in the given function we get,
i.e.,
i.e.,
Therefore, minimum value of f(x) is 42.
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