Saved Bookmarks
| 1. |
Draw the graph of the function f(x) = (1/x)^(x) |
|
Answer» Solution :We have `f(x) = (1/x)^(x)` Clearly, DOMAIN of the function is `x gt 0` DIFFERENTIATING we get, `f^(')(x)=(1/x)^(x)(log1/x-1)` `f^(')(x) =0 rArr log1/x=1 = log_(e)e rArr 1/x=e rArr x=1/e` Also for `x lt 1//e, f^(')(x)` is positive and for `x gt 1//e, f^(')(x)` is negative. Therefore, the maximum value of the function is `e^(1//e)`. Also, `UNDERSET(x to 0)"lim"(1/x)^(x) = e^(underset(xto0)"lim"xlog(1/x))=e^(underset(xto0)"-lim"xlogx)=e^(underset(xto0)-"lim"(logx)/(1/x))=e^(underset(xto0)-"lim"(1/x)/(-1/x^(2)))=e^(0)=1` `underset(xto infty)"lim"(1/x)^(x)=0` From the above discussion, the graph of the function is as shown in the following figure.
|
|