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.

For the function f(x) = x^(x) , find the points at which it is (i) increasing and (ii) decreasing. Hence determine which of e^(pi), pi^(e) is greater.

f(x)=x.e^(-x) is decreasing in

Show that the function given by f (x) = e ^(2x) is increasing on R.

Show that Lt_(xto0)(e^(x)-1)/x=1

Show that Lt_(x to 0)(e^(x)-1)/(x)=1

Statement-1: The fucntion x^(2)(e^(x)+e^(-x)) is increasing for all x gt 0 . Statement-2 : The function x^(2)e^(x) and x^(2)e^(-x) are increasing for all x gt 0 and the sum of two increasing functions in any interval (a,b) is an increasing function in (a,b)

Show that int x ^(n) . e ^(-x) dx= -x^(n) e ^(-x) +n int x ^(n-1) . e ^(-x) dx

Use the function f(x)=x^((1)/(x))(xgt0) to ascertain whether p^(e) or e^(p) is greater .

int e^(x//2) sin ((pi)/(4)+ (pi)/(2) ) dx =