SATHEE: Application of Derivatives 4 Question 24

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) $1 / 4$

(d) $1 / 3$

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Answer:

Correct Answer: 25. (b)

Solution:

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

$\begin{aligned} \Rightarrow f^{'} (x) & = 25 x^{24} (1 - x)^{75} - 75 x^{25} (1 - x)^{74} \\ = 25 x^{24} (1 - x)^{74} [(1 - x) - 3 x] \\ = 25 x^{24} (1 - x)^{74} (1 - 4 x) \end{aligned}$

For maximum value of $f (x)$ , put $f^{'} (x) = 0$

$\begin{array}{ll} \Rightarrow & 25 x^{24} (1 - x)^{74} (1 - 4 x) = 0 \\ \Rightarrow & x = 0, 1, \frac{1}{4} \\ Also, at & x = 0, y = 0 \\ At & x = 1, y = 0 \\ and at & x = 1 / 4, y > 0 \end{array}$

$∴ f (x)$ attains maximum at $x = 1 / 4$ .

Application of Derivatives 4 Question 24

Answer:

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Application of Derivatives 4 Question 24

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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