# How to find maxima and minima of a function involving a factorial

i need to find the value of y when the bell curve for the following function reaches its maximum , i can solve the problem easily on a MAS software but i needed to know a more mathematical approach . here's the simplified equation

$$\frac{3^{-y}}{(25-y)! y!}$$

and here's the original equation which i need to solve

$$\frac{\left(\frac{3}{4}\right)^{25} 25!}{3^y (25-y)! y!}$$

in a nutshell i need to find the value of y when the above function reaches its maxima

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HINT:

If $T_y=\frac{\left(\frac34\right)^{25}25!}{3^y(25-y)! y!}$

Clearly, $0\le y\le 25$

$$\frac{T_{y+1}}{T_y}=\frac{3^y(25-y)!y!}{3^{y+1}(25-y-1)!(y+1)!}=\frac{25-y}{3(y+1)}$$

Check when $\frac{T_{y+1}}{T_y}>\text{ or } = \text{ or } <1$

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The pattern is like : askiitians.com/iit-jee-algebra/… – lab bhattacharjee Jul 6 '13 at 16:08
Intuitively, why does comparing it to 1 help get you the answer? Does it have to do with the Ratio test to see if a series absolutely converges? – dmonopoly Sep 15 '13 at 20:36
@dmonopoly, as each term is positive and $a>b>0\iff \frac ab>1$ – lab bhattacharjee Sep 16 '13 at 14:00