If `f^(prime)x=g(x)(x-a)^2,w h e r eg(a)!=0,a n dg` is continuous at `x=a ,` then `f` is increasing in the neighbourhood of `a` if `g(a)>0` `f` is increasing in the neighbourhood of `a` if `g(a)<0` `f` is decreasing in the neighbourhood of `a` if `g(a)>0` `f` is decreasing in the neighbourhood of `a` if `g(a)<0`

If f(x)=x/(sinx)a n dg(x)=x/(tanx),w h e r e0ltxlt=1, then in this interval

If g(x) =2f(2x^3-3x^2)+f(6x^2-4x^3-3) AA x in R and f''(x) gt 0 AA x in R then g(x) is increasing in the interval

Statement 1: The function f(x)=x In x is increasing in (1/e ,oo) Statement 2: If both f(x)a n dg(x) are increasing in (a , b),t h e nf(x)g(x) must be increasing in (a,b).

If f(x)=a x^2+b x+c ,g(x)=-a x^2+b x+c ,w h e r ea c!=0, then prove that f(x)g(x)=0 has at least two real roots.

Which of the following statement is always true? (a)If f(x) is increasing, then f^(-1)(x) is decreasing. (b)If f(x) is increasing, then 1/(f(x)) is also increasing. (c)If fa n dg are positive functions and f is increasing and g is decreasing, then f/g is a decreasing function. (d)If fa n dg are positive functions and f is decreasing and g is increasing, the f/g is a decreasing function.

If f''(x) gt forall in R, f(3)=0 and g(x) =f(tan^(2)x-2tanx+4y)0ltxlt(pi)/(2) ,then g(x) is increasing in

If f(x)=(x+1)/(x-1)a n dg(x)=1/(x-2),t h e n discuss the continuity of f(x),g(x),a n dfog(x)dot

If f(x)=|x-2|a n dg(x)=f[f(x)],t h e ng^(prime)(x)= ______ for x>2

If fogoh(x) is an increasing function, then which of the following is possible? (a) f(x),g(x),a n dh(x) are increasing (b) f(x)a n d h(x) are increasing and g(x) is decreasing (c) f(x),g(x),a n dh(x) are decreasing

Let g(x)=f(x)+f(1-x) and f "(x)>0AAx in (0,1)dot Find the intervals of increase and decrease of g(x)dot