In X-Y plane, the path defined by the equation `(1)/(x^(m))+(1)/(y^(m)) +(k)/((x+y)^(n)) =0`, is a pair of lines if ` m=k=−1,n=1`

Explanation

`f(x)=(1/x)^(x)`
`f(x) =(1/x)^(x)(log_(e)(1))/(x)-1`
`f(x) 0 rarr log_(e)(1)/(x)=1 rArr x=(1)/(c )`
Sign scheme of f(x) is as follows:

From the sing scheme `x=(1)/(c )` is point of maxima.
Also `underset(xrarr0)lim(1/x)^(x)=e^(xrarr0^(limxlog))(1/x)=e^(xrarr0^(-lixxlogx))=e(0)=1`
and `underset(xrarr00)lim (1/x^(x))=0`
so, graph of the function is as shown in the following

From the graph range of the function is `0,f(1//e) or 0,e^(1)`
Now `pigte`
`f(pi)ltf(e)`
`(1)/(pi^(pi))lt(1)/(e^(e))`
`(1)/(pi^(1/e)) lt (1/e^(1))/(pi)`