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Please solve RD Sharma class 12 chapter 16 Increasing and Decreasing Function Excercise Fill in the blanks Question 2: Maths Textbook Solution.

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Answer: The is function decreasing in the intervals \left ( 1,\infty \right )

Hint: The function is decreasing if f'\left ( x \right )<0

Given: f\left ( x \right )=\frac{2x^{2}-1}{x^{4}}

Explanation: We have,

f\left ( x \right )=\frac{2x^{2}-1}{x^{4}},x>0                                                        ….(i)

Differentiate (i) with respect to x

f\left ( x \right )=\frac{x^{4}\left ( 2.2x \right )-\left ( 2x^{2}-1 \right )4x^{3}}{\left ( x^{4}\right )^{2}}







So,  \frac{4(1-x)(1+x))}{x^{5}}<0                         [  \because f (x) is decreasing]

\Rightarrow \frac{(1-x)(1+x)}{x^{5}}<0             

\Rightarrow (1-x)(1+x)<0

\Rightarrow (1-x)<0                                  [ as (1+x)<0\Rightarrow x<-1 is not possible.since x>0 is given ]

\Rightarrow x>1

\therefore x\epsilon (1, \infty )

Thus, the function decreases in the interval (1, \infty )

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