Discussion Forum



1)

 On the interval [0,1] , the function x25 (1-x)75 takes its maximum value at the point


A) 0

B) $\frac{1}{4}$

C) $\frac{1}{2}$

D) $\frac{1}{3}$

Answer:

Option B

Explanation:

 Let  $f(x)=x^{25}(1-x)^{75},x\in[0,1]$

$\Rightarrow f'(x)=25x^{24}(1-x)^{75}-75x^{25}(1-x)^{74}$

$=25x^{24}(1-x)^{74}\left\{(1-x)-3x\right\}$

$=25x^{24}(1-x)^{74}\left\{(1-4x)\right\}$

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 We can see that f'(x)  is positive  for $x < \frac{1}{4}$

 and f '(x) is negative for $x > \frac{1}{4}$

 Hence, f(x) attains maximum at x = $\frac{1}{4}$