Let `g^(prime)(x)>0a n df^(prime)(x)<0AAx in Rdot` Then `(f(x+1))>g(f(x-1))` `f(g(x-1))>f(g(x+1))` `g(f(x+1))

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: R-> be a differentiable function AAx in R . If the tangent drawn to the curve at any point x in (a , b) always lies below the curve, then (a) f^(prime)(x)<0,f^(x)<0AAx in (a , b) (b) f^(prime)(x)>0,f^(x)>0AAx in (a , b) (c) f^(prime)(x)>0,f^(x)>0AAx in (a , b) (d) non eoft h e s e

Let f'(x) gt0 and g'(x) lt 0 " for all " x in R Then

Suppose that f(x) is differentiable invertible function f^(prime)(x)!=0a n dh^(prime)(x)=f(x)dot Given that f(1)=f^(prime)(1)=1,h(1)=0 and g(x) is inverse of f(x) . Let G(x)=x^2g(x)-x h(g(x))AAx in Rdot Which of the following is/are correct? G^(prime)(1)=2 b. G^(prime)(1)=3 c. G^(1)=2 d. G^(1)=3

let ( f(x) = 1-|x| , |x| 1 ) g(x)=f(x+1)+f(x-1)

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 int(f^(prime)(x)g(x)-g^(prime)(x)f(x))/((f(x)+g(x))sqrt(f(x)g(x)-g^2(x)))d x=sqrt(m)tan^(- 1)(sqrt((f(x)-g(x))/(ng(x))+C) where m,n in N and 'C' is constant of integration (g(x) > 0). Find the value

Suppose fa n dg are functions having second derivative f'' and g' ' everywhere. If f(x)dotg(x)=1 for all xa n df^(prime)a n dg' are never zero, then (f^('')(x))/(f^(prime)(x))-(g^('')(x))/(g^(prime)(x))e q u a l (a)(-2f^(prime)(x))/f (b) (2g^(prime)(x))/(g(x)) (c)(-f^(prime)(x))/(f(x)) (d) (2f^(prime)(x))/(f(x))

If f(1) =g(1)=2 , then lim_(xrarr1) (f(1)g(x)-f(x)g(1)-f(1)+g(1))/(f(x)-g(x)) is equal to