Originally Posted by

**lptuyen** The problem is:

Show that the function $\displaystyle f(K) = \sum\limits_{i = 1}^K \binom{K}{i} {x^{K - i}{y^i}}$ is a monotonically decreasing function when $\displaystyle x+y \le 0.5$, $\displaystyle x \ge 0$ and$\displaystyle y \ge 0$, $\displaystyle K $ is a positive integer number.

A proof is as follows (but I cannot understand):

Consider the function $\displaystyle f(t)$, with $\displaystyle t \in {R^ + }$. The first order derivative of $\displaystyle f(t)$ with respect to$\displaystyle t$ is:

$\displaystyle f{(t)^'} = \ln (x + y){x^t}\{ {[{x^{ - 1}}{(x + y)^t} - \ln (x)\ln (x + y)]^{ - 1}}\}$

Because $\displaystyle {[{x^{ - 1}}{(x + y)^t} - \ln (x)\ln (x + y)]^{ - 1}} \ge 0$ when $\displaystyle x+y \le 0.5$, so $\displaystyle f{(t)^'} \le 0$, and finally, we obtain the function $\displaystyle f(t)$ is monotonically decreasing function. The function $\displaystyle f(K)$ is a special case of $\displaystyle f(t)$, and we have the proof for the problem.

The above proof is very short, and I cannot understand that. Can everyone give me some ideas?. Thanks in advance.