question_answer

If \[{{x}^{2}}+9<{{(x+3)}^{2}}<8x+25,\]then f(x) is

A)  increasing on \[\frac{x+y}{x-y}=\frac{5}{2},\]

B) decreasing on R

C)  increasing on R  

D) decreasing on \[\frac{x}{y}\]

Correct Answer: A

Solution :

 We have,\[a{{x}^{2}}+2hxy+b{{y}^{2}}=1,a>0\] On differentiating both sides w.r.t. x, we get \[\frac{\pi }{4}\] \[f(x)=x{{e}^{-x}}\]\[[0,\infty ),\] \[0\]\[\frac{1}{e}\] Since, \[\theta \] for all x. Therefore, signs of f'(x) for different values of x are as shown below: Clearly, f{x) is increasing on \[\frac{\pi }{6}\] and decreasing on\[\frac{\pi }{4}\]