question_answer

    If a and b are two different positive real numbers, then which of the following statements is true?

A)  \[2\sqrt{ab}>a+b\]        

B)  \[2\sqrt{ab}<a+b\]

C)  \[2\sqrt{ab}=a+b\]        

D)  None of these

Correct Answer: B

Solution :

                The area bounded by the given curves is shown in figure. The required area is \[=\int_{x=-1}^{0}{\{(3+x)-(-x+1)\}}dx\]                                 \[+\int_{x=0}^{0}{\{(3-x)-(-x+1)\}}dx\]                                 \[+\int_{x=1}^{2}{\{(3-x)-(x-1)\}}dx\] \[=\int_{-1}^{2}{(3x+2)dx}+\int_{0}^{1}{2dx}+\int_{1}^{2}{(4-2x)}dx\] \[=({{x}^{2}}+2x)_{-1}^{0}+(2x)_{0}^{1}+(4x-{{x}^{2}})_{1}^{2}\] \[=0-(1-2)+(2-0)+(8-4-4+1)\] \[=1+2+1\] \[=4\]