prove that `(x)/(1+x) lt log (1+x)ltx`, for all `xgt 0`

Prove that ln(1+x) 0.

Prove that log_(e)(1+x)ltxforxgt0

If y log x= x-y , prove that (dy)/(dx)= (log x)/((1+log x)^(2))

Which of the following is not true log(1+x)ltx for xgt0

For x in (0,1) Prove that x-(x^(3))/(3) lt tan^(-1) x lt x -(x^(3))/(6) hence or otherwise find lim_( x to0) [(tan^(-1)x)/(x)]

Prove that ln (1+1/x) gt (1)/(1+x), x gt 0 . Hence, show that the function f(x)=(1+1/x)^(x) strictly increases in (0, oo) .

If 0ltxlt1 , prove that: log(1+x)+log(1+x^2)+log(1+x^4)+… oo=-log(1-x)

Show that 1+x ln (x+sqrt(x^2+1))geqsqrt(1+x^2) for all xgeq0.

0 lt a lt 2 pi , sin^(-1)(sin a ) lt x^(2)-2x for all x in I then a in

Let A={x in R :0lt=xlt=1}dot If f: A->A is defined by f(x)={(x ,, if x in Q) , (1-x ,, if x!in Q):} then prove that fof(x)=x for all x in Adot