If`f(x)a n dg(x)` be two function which are defined and differentiable for all `xgeqx_0dot` If `f(x_0)=g(x_0)a n df^(prime)(x)>g^(prime)(x)` for all `f> x_0,` then prove that `f(x)>g(x)` for all `x > x_0dot`

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 f(x) and g(x) be defined and differntiable for all x ge x_0 and f(x_0)=g(x_0) f(x) ge (x) for x gt x_0 then

If f(x)=(a-x^(n))^((1)/(n)),a>0 and n in N, then prove that f(f(x))=x for all x.

Let g(x) be the inverse of an invertible function f(x), which is differentiable for all x, then g'f(x) is equal to

Let the function f satisfies f(x)*f^(prime)(-x)=f(-x)*f^(prime)(x) foe all x and f(0)=3 The value of f(x).f^(prime)(-x)=f(-x).f^(prime)(x) for all x, is

Given that f(x) gt g(x) for all x in R and f(0) =g(0) then

Let f:R rarr R be a twice differentiable function such that f(x+pi)=f(x) and f'(x)+f(x)>=0 for all x in R. show that f(x)>=0 for all x in R .

Let f be a differentiable function satisfying f(x/y)=f(x)-f(y) for all x ,\ y > 0. If f^(prime)(1)=1 then find f(x)dot

Let g(x) be the inverse of an invertible function f(x), which is differentiable for all real xdot Then g^('')(f(x)) equals. (a) -(f^('')(x))/((f^'(x))^3) (b) (f^(prime)(x)f^('')(x)-(f^(prime)(x))^3)/(f^(prime)(x)) (c) (f^(prime)(x)f^('')(x)-(f^(prime)(x))^2)/((f^(prime)(x))^2) (d) none of these

Let f(x)>0 and g(x)>0 with f(x)>g(x)AA x in R are differentiable functions satisfying the conditions (i) f(0)=2,g(0)=1 , (ii) f'(x)=g(x) , (iii) g'(x)=f(x) then