IIT-JEE 1995 ScreeningQuestion 1
On the interval $[0,1]$ the function $x^{25}(1-x)^{75}$ takes its maximum value at the point
Differentiation and its Application
Solution
Concept. Factor $f'$ and find its sign change inside $(0,1)$. Formulas used. (1) Product ruleStep 1 - Derivative. By (1), $$f'(x)=x^{24}(1-x)^{74}\big[25(1-x)-75x\big]=x^{24}(1-x)^{74}(25-100x).$$ Step 2 - Sign. Positive for $x<\frac14$, negative for $x>\frac14$. Also $f(0)=f(1)=0$. Step 3 - Conclude. Maximum at $x=\frac14$. Answer: (B)