Application of Derivatives 4 Question 24
25. On the interval $[0,1]$, the function $x^{25}(1-x)^{75}$ takes its maximum value at the point
(1995, 1M)
(a) 0
(b) $1 / 4$
(c) $1 / 2$
(d) $1 / 3$
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Answer:
Correct Answer: 25. (b)
Solution:
- Let $f(x)=x^{25}(1-x)^{75}, x \in[0,1]$
$$ \begin{aligned} \Rightarrow \quad f^{\prime}(x) & =25 x^{24}(1-x)^{75}-75 x^{25}(1-x)^{74} \\ & =25 x^{24}(1-x)^{74}[(1-x)-3 x] \\ & =25 x^{24}(1-x)^{74}(1-4 x) \end{aligned} $$
For maximum value of $f(x)$, put $f^{\prime}(x)=0$
$$ \begin{array}{ll} \Rightarrow & 25 x^{24}(1-x)^{74}(1-4 x)=0 \\ \Rightarrow & x=0,1, \frac{1}{4} \\ \text { Also, at } & x=0, y=0 \\ \text { At } & x=1, y=0 \\ \text { and at } & x=1 / 4, y>0 \end{array} $$
$\therefore f(x)$ attains maximum at $x=1 / 4$.
