A) \[{{\left( \frac{\pi }{4} \right)}^{1/\sqrt{2}}}\left( \frac{\sqrt{2}}{2}\log \frac{4}{\pi }-\frac{2\sqrt{2}}{\pi } \right)\]
B) \[{{\left( \frac{\pi }{4} \right)}^{1/\sqrt{2}}}\left( \frac{\sqrt{2}}{2}\log \frac{4}{\pi }+\frac{2\sqrt{2}}{\pi } \right)\]
C) \[{{\left( \frac{\pi }{4} \right)}^{1/\sqrt{2}}}\left( \frac{\sqrt{2}}{2}\log \frac{\pi }{4}-\frac{2\sqrt{2}}{\pi } \right)\]
D) \[\left( \frac{\sqrt{2}}{2}\log \frac{\pi }{4}+\frac{2\sqrt{2}}{\pi } \right)\]
Correct Answer: A
Solution :
Since, \[f(x)=|x{{|}^{\sin x}}\] when\[x<0\] \[f(x)={{(-x)}^{(-\sin x)}}\] \[={{e}^{-\sin x\log (-x)}}\] \[\Rightarrow \] \[f(x)={{e}^{-\sin x\log (-x)}}\] \[\left( -\cos x\log (-x)-\frac{\sin x}{x} \right)\] \[\Rightarrow \] \[f(x)={{(-x)}^{-\sin x}}\] \[\left( -\cos x\cdot \log (-x)-\frac{\sin x}{x} \right)\] \[\Rightarrow \] \[f\left( -\frac{\pi }{4} \right)={{\left( \frac{\pi }{4} \right)}^{1/\sqrt{2}}}\] \[\left( -\frac{1}{\sqrt{2}}\log \frac{\pi }{4}+\frac{4}{\pi }\left( -\frac{1}{\sqrt{2}} \right) \right)\] \[={{\left( \frac{\pi }{4} \right)}^{1/\sqrt{2}}}\left( \frac{\sqrt{2}}{2}\log \frac{4}{\pi }-\frac{2\sqrt{2}}{\pi } \right)\]
You need to login to perform this action.
You will be redirected in
3 sec
