Let `g(x)gt0andf'(x)lt0,AAx in R,` then show
`g(f(x+1))ltg(f(x-1))`
`f(g(x+1))ltf(g(x-1))`

Let g'(x)gt 0 and f'(x) lt 0 AA x in R , then

Let f'(x)gt0andf''(x)gt0 where x_(1)ltx_(2). Then show f((x_(1)+x_(2))/(2))lt(f(x_(1))+(x_(2)))/(2).

If f(x) is monotonically increasing function for all x in R, such that f''(x)gt0andf^(-1)(x) exists, then prove that (f^(-1)(x_(1))+f^(-1)(x_(2))+f^(-1)(x_(3)))/(3)lt((f^(-1)(x_(1)+x_(2)+x_3))/(3))

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.

Let g(x)=f(x)+f(1-x) and f''(x)<0 , when x in (0,1) . Then f(x) is

Let f and g be two differentiable functions on R such that f'(x)>0 and g′(x) g(f(x-1)) (b) f(g(x))>f(g(x+1)) (c) g(f(x+1))

Let f and g be differentiable functions such that: xg(f(x))f\'(g(x))g\'(x)=f(g(x))g\'(f(x))f\'(x) AA x in R Also, f(x)gt0 and g(x)gt0 AA x in R int_0^xf(g(t))dt=1-e^(-2x)/2, AA x in R and g(f(0))=1, h(x)=g(f(x))/f(g(x)) AA x in R Now answer the question: f(g(0))+g(f(0))= (A) 1 (B) 2 (C) 3 (D) 4

Suppose f and g are differentiabel functions such that xg (f(x))f'(g (x))g '(x) =f(g(x))g '(f(x)) f'(x) AA x in R and f is positive AA n in R. Also int _( 0)^(x) f (g(t )) dt =1/2 (1-e^(-2x))AA x in R, g (f(0))=1 and h (x) = (g(f (x)))/(f (g(x)))AA x in R. The graph of y =h (x) is symmetric with respect to line:

Let phi(x,t)={(x(t-1),xlet),(t(x-1), tltx):} , where t is a continuous function of x in [0,1] . Let g(x)=int_0^1 f(t)phi(x,t)dt , then g\'\'(x) = (A) g(0)+g(1)=1 (B) g(0)=0 (C) g(1)=1 (D) g\'\'(x)=f(x)

Let f(x) be a function such that f(x).f(y)=f(x+y) , f(0)=1 , f(1)=4 . If 2g(x)=f(x).(1-g(x))