Show that `(x)/((1+x))ltlog(1+x)ltx" for "xgt0.`

Show that cot^(-1)x={(tan^(-1)(1//x),xgt0),(pi+tan^(-1)(1//x),xlt0):}

Consider the fucntion f(x)=((1)/(x))^(2x^2) , where xgt0 . At what value of x does the function attain maximum value ?

IF f(x)=1/(x+1)-log(1+x),xgt0 , then f is

f(x) = {(1-2x", for " x le0),(1+2x", for "xgt0):} then int_(-1) ^(1) f(x) dx=

Determine the values of 'a' and 'b' such that the following function is continuous at x = 0 : f(x)={{:((x+sinx)/(sin(a+1)x)", if "-piltxlt0),(2", if "x=0),(2(e^(sinbx)-1)/(bx)", if "xgt0):} .

Let f be differentiable function such that f'(x)=7-3/4(f(x))/x,(xgt0) and f(1)ne4" Then " lim_(xto0^+) xf(1/x)

Consider the following statement 1. f(x)=e^(x) , where xgt0 2. g(x)=|x-3| which of the above function is/are continuous?

The least value of alpha in R for which 4alphax^(2)+(1)/(x)ge1, for all xgt0, is