`f(x)=tan^(-1)(sinx+cosx), x gt0` is always and increasing function on the interval

Show that the function f given by f(x)=tan^(-1)(sin x + cos x), x gt 0 is always an increasing function in (0, (pi)/(4)) .

Show that f(x)=tan^(-1)(sin x + cos x) is an increasing function in (0, (pi)/(4)) .

Without using differentiation, prove that f(x)=ax+b , where a gt 0 is strictly increasing function AA x in R .

For the fucntion f(x)=x cos ""1/x, x ge 1 which one of the following is incorrect ? (a)for atleast one x in the interval [1, oo), f(x + 2) - f(x) lt 2 (b) underset(x rarr oo)(lim) f'(x) = 1 (c)for all x in the interval [1, oo), f(x+2) - f(x) gt 2 (d)f'(x) is strictly decreasing in the interval [1, oo)

The function f(x)=(2x^(2)-1)/(x^(4)), x gt 0 , decreases in the interval …………. .

Let f be a function defined on [a,b] such that f'(x)gt 0 , for all x in (a, b) . Then prove that f is an increasing function on (a, b).

f(x)=(x)/(log_(x)e) is increasing on the interval ……, where x inR^(+)-{1} .

If f(x)=((1-tanx)/(1+sinx))^(cosec x) is to be made continuous at x=0, then f(0) must be equal to

The function f (x) = tan x-x ………..

Find the intervals in which the function f given f(x)=x^(3)+(1)/(x^(3)), x ne 0 is (i) increasing (ii) decreasing.