If `f(x)=x/(sinx) \ a n d \ g(x)=x/(tanx),w h e r e \ 0ltxlt=1,` then in this interval

Let f'(sinx)lt0andf''(sinx)gt0,AAx in (0,(pi)/(2)) and g(x)=f(sinx)+f(cosx), then find the interval in which g(x) is increasing and decreasing.

Consider f, g and h be three real valued function defined on R. Let f(x)=sin3x+cosx,g(x)=cos3x+sinx and h(x)=f^(2)(x)+g^(2)(x). Then, The length of a longest interval in which the function h(x) is increasing, is

If f(x)=sinx,g(x)=x^(2)andh(x)=logx. IF F(x)=h(f(g(x))), then F'(x) is

If f(x)=x+tanx and g(x) is the inverse of f(x) , then differentiation of g(x) is (a) 1/(1+[g(x)-x]^2) (b) 1/(2-[g(x)+x]^2) (c)1/(2+[g(x)-x]^2) (d) none of these

For x in R, f(x) =|log_(e) 2-sinx| and g(x) = f(f(x)) , then

f(x)=e^(lncot), g(x)=cot^(- 1)x

Let f(x)=e^({e^(|x|sgnx)})a n dg(x)=e^([e^(|x|sgnx)]),x in R , where { } and [ ] denote the fractional and integral part functions, respectively. Also, h(x)=log(f(x))+log(g(x)) . Then for real x , h(x) is

If the functions f(x)=x^(3)+e^(x//2) " and " g(x)=f^(-1)(x) , the value of g'(1) is ………… .

Let f:[1/2,1]->R (the set of all real numbers) be a positive, non-constant, and differentiable function such that f^(prime)(x)<2f(x))a n df(1/2)=1 . Then the value of int_(1/2)^1f(x)dx lies in the interval (a) (2e-1,2e) (b) (3-1,2e-1) ((e-1)/2,e-1) (d) (0,(e-1)/2)

If a lt 0 and f(x)=e^(ax )+ e^(-ax) is monotonically decreasing . Find the interval to which x belongs.