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Please solve RD Sharma class 12 chapter 16 Increasing and Decreasing Function Excercise Fill in the blanks Question 10: Maths Textbook Solution.

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Answer: The largest interval in which f(x) is increasing is 

Hint: Function is increasing if f'(x)>0 .

Given: f(x)=x^{1/x}

Explanation: We have,


Taking logarithm  on both sides,

\log f(x)=\frac{1}{x}\log x                                             …(i)

Differentiate (i) with respect to x

\frac{1}{f(x)}f'(x)=\frac{1}{x}\cdot \frac{1}{x}+\log x\cdot \left ( -\frac{1}{x^{2}} \right )

\therefore f'(x)=\left ( \frac{1}{x^{2}}-\frac{1}{x^{2}}\log x \right )\cdot x^{1/x}                               \left [ \because f(x)=x^{\frac{1}{x}} \right ]

f'(x)=\frac{1}{x^{2}}(1-\log x)x^{\frac{1}{x}}

\because f(x)  is strictly increasing,


\Rightarrow \frac{1}{x^{2}}(1-\log x)x^{1/x}>0

Now, if  1-\log x>0

\Rightarrow \log x<1

\Rightarrow x<e


So, the largest interval in which f(x)  in increasing in (1,e )

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