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Which one is the biggest value? Help

As I just have my test. I found this kind of problem :
Which one of these have a biggest value?
A. \(99^{100}\)
B. \(98^{101}\)
C. \(105^{97}\)
D. \(101^{98}\)
E. \(100^{99}\)

I will appreciate any feedbacks. Thanks.

Note by Jansen Wu
1 year ago

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Hint:- Look at the function \( f(x) = x^{199 - x} \) Siddhartha Srivastava · 1 year ago

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@Siddhartha Srivastava so how can i use that function to find it ? Jansen Wu · 1 year ago

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@Jansen Wu \( f'(x) = x^{199 - x}(-\ln{x} + \frac{109 -x}{x}) \).

In [98,105], we have \( \ln{x} > 1 > \frac{109-x}{x} \). So \( f'(x) < 0 \), which means that \( f(x) \) decreasing in [98,105]. So the maximum value occurs at the smallest \( x \), i.e., 98^101. Siddhartha Srivastava · 1 year ago

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@Siddhartha Srivastava means that the answer for this kind of problem is \(98^{101}\) then thx to you much... Jansen Wu · 1 year ago

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