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Statement 1 : The function x^{2}(e^ {x}+e^{-x}) is increasing for all x> 0.

Statement 2 : The functions x^{2}e^{x} and x^{2}e^{-x} are increasing for all x> 0 and the sum of two increasing functions in any interval (a,b) is an increasing function in (a,b)

Option: 1

Statement 1 is false ; Statement 2 is true


Option: 2

Statement 1 is true ; Statement 2 is true ; Statement 2 is not correct explanation for Statement 1


Option: 3

Statement 1 is true ; Statement 2 is false


Option: 4

Statement 1 is true ; Statement 2 is true ; Statement 2 is a correct explanation for Statement 1


Answers (1)

best_answer

\\\text{Let } y=x^{2} \cdot e^{-x} \\\text{For increasing function. }\\\frac{d y}{d x}>0 \Rightarrow x\left[(2-x) e^{-x}\right]>0\\ \because x>0, \therefore(2-x) e^{-x}>0 \\\Rightarrow(2-x) \frac{1}{e^{x}}>0\\ \text{For }0<x<2,(2-x)<0\\ \therefore \frac{1}{e^{x}}<0,\text{ but it is not possible}

Hence, statement 2 is false.

Posted by

Divya Prakash Singh

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