Discussion Forum

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

Answer:

Explanation:

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

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

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

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

 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 = $\frac{1}{4}$