Let `f(x)a n dg(x)` be two function having finite nonzero third-order derivatives `f^(x)a n dg^(x)` for all `x in Rdot` If `f(x)g(x)=1` for all `x in R ,` then prove that `f^(/)f^(prime)-g^(/)g^(prime)=3(f^(/)f-g^(/)g)dot`

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))

Let f(x) and g(x) be two functions which are defined and differentiable for all x>=x_(0). If f(x_(0))=g(x_(0)) and f'(x)>g'(x) for all x>x_(0), then prove that f(x)>g(x) for all x>x_(0).

Let g(x) =f(x)-2{f(x)}^2+9{f(x)}^3 for all x in R Then

If f(x)=log_(e)x and g(x)=e^(x) , then prove that : f(g(x)}=g{f(x)}

Let f and g be two functions from R into R defined by f(x) =|x|+x and g(x) = x " for all " x in R . Find f o g and g o f

Iff (x) and g(x) be two function which are defined and differentiable for all x>=x_(0). If f(x_(0))=g(x_(0)) and f'(x)>g'(x) for all f>x_(0), then prove that f(x)>g(x) for all x>x_(0).

If g is inverse of f then prove that f''(g(x))=-g''(x)(f'(g(x)))^(3).

Suppose f and g are functions having second derivatives f' and g' every where,if f(x)*g(x)=1 for all x and f'',g'' are never zero then (f'(x))/(f'(x))-(g'(x))/(g'(x)) equals

Let f(x) and g(x) be functions which take integers as arguments.Let f(x+y)=f(x)+g(y)+8 for all intege x and y.Let f(x)=x for all negative integers x and let g(8)=17. Find f(0) .