At what integral value of x , function f(x)= C(100,x)*(2^x) attains maximum , C(100,x) denotes combination of 100 different things taken x at a time.

Question

I tried it by taking derivative and putting it equal to zero (for obtaining Maxima or minima), but I don't know how to take derivative of C(100,x). Infact derivative of C (100,x) doesn't exist,so how to start?

Answer 1

For maxima calculate range of x such that $$\frac{f(x+1)}{f(x)}<1$$ $$\frac{2(100-x)}{x+1}<1$$ $$x>\frac{199}{3}$$ Thus for $x=67$, $f(x)$ is maximum. Similarly you can find minimum.