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
2 years, 7 months ago

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Hint:- Look at the function $$f(x) = x^{199 - x}$$

- 2 years, 7 months ago

so how can i use that function to find it ?

- 2 years, 7 months ago

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

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.

- 2 years, 7 months ago

means that the answer for this kind of problem is $$98^{101}$$ then thx to you much...

- 2 years, 7 months ago