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Find the point of extream of the function f(x) =(12)/(x^(x)).Draw the graph of the function and hence find therange of function .Also detrmine which is bigger,(1)/(pi)^(1/e) or (1)/(e )^((1)/(pi))? |
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Answer» Solution :`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 isas 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)` |
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