Show that `x + (1)/(x)` is increasing on `[1, oo)`.

Show that x^(1//x) is increasing in (0,e) and decreasing in (e,oo) . Hence determine which of e^pi,pi^e is greater.

If x gt 0 , show that log(1+x) - (2x)/(2+x) is increasing.

If f(x) = {(3x^(2) + 12x - 1, -1 le x le 2),(" "37-x, 2 lt x le 3):} , show that f(x) is increasing in [-1, 2]

Show that f(x)=|x| is strictly decreasing on (-oo, 0) and strictly increasing on (0, oo) .

Show that the functions ln""((1+x))/(x) and (x)/((1+x)ln(1+x)) are decreasing on (0, oo) .

int (log (1+x))/(1+x) dx on (-1, oo).

Obsereve the following the lists {:("List-I","List-II"),("A) "f(x)=x^2-2x+5" is increasing in","1) "xgt1/e),("B) "f(x)=x^x" decreases for","2) "(-oo,-1)uu(1,oo)),("C) "f(x)=x+1/x" is increasing in","3) "(1,oo)),("D) "f(x)=9-6x-2x^2-x^3" is decreasing for","4) "(-oo,oo)),("","5) "0ltxlt1/e):}

{:("List-I","List -II"),((A)f(x)=x^(2)-2x+5 "is increasing in","P"phi),((B)f(x)=e^(-x) "is increasing in ","q" (-oo,1)uu(2,oo)),((C)f(x)=logx "is increasing in","r" R),((D)f(x)=(x^(3))/(3)-(3x^(2))/(2)+2x+5 " is increasing in ","s"(0,oo)),(,t(1,oo)):}

Evaluate the integerals. int (1)/(x log x) dx on (-1, oo)