Use the function `f(x)=x^(1/x),x >0,` to determine the bigger of the two numbers `e^(pi)a n dpi^edot`

The function f(x)=2sin x-e^(x), AA x in [0, pi] has

Prove that inequality e^(x)>(1+x) for all x in R_(0) and use it to determine which of the two numbers e^(pi) and pi' is greater.

Consider the function f(x)=e^(-2x)sin2x over the interval (0,pi/2) . A real number c in(0,pi/2) , as guaranteed by Rolle's theorem such that f'(c)=0, is

The function f(x)=(log(pi+x))/(log(e+x)) is

For the functions f(x)= int_(0)^(x) (sin t)/t dt where x gt 0 . At x=n pi f(x) attains

If f(x)=e^(x) sin x, x in [0, pi] , then

The function f(x)=x*e^(-((1)/(|x|)+(1)/(x))) if x!=0 and f(x)=0 if x=0 then