Let `f(x)a n dg(x)` be two functions 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 `x > x_0,` then prove that `f(x)>g(x)` for all `x > x_0dot`

Let fa n dg be differentiable on R and suppose f(0)=g(0)a n df^(prime)(x)lt=g^(prime)(x) for all xgeq0. Then show that f(x)lt=g(x) for all xgeq0.

Let f be differentiable for all x , If f(1)=-2a n df^(prime)(x)geq2 for all x in [1,6], then find the range of values of f(6)dot

Let f(x)a n dg(x) be two differentiable functions in Ra n df(2)=8,g(2)=0,f(4)=10 ,a n dg(4)=8. Then prove that g^(prime)(x)=4f^(prime)(x) for at least one x in (2,4)dot

Let f(x)a n dg(x) be two continuous functions defined from RvecR , such that f(x_1)>f(x_2)a n dg(x_1) f(g(3alpha-4))

Let f(x+y)=f(x)dotf(y) for all xa n dydot Suppose f(5)=2a n df^(prime)(0)=3. Find f^(prime)(5)dot

If function f satisfies the relation f(x)*f^(prime)(-x)=f(-x)*f^(prime)(x) for all x ,

Let f(x0 be a non-constant thrice differentiable function defined on (-oo,oo) such that f(x)=f(6-x)a n df^(prime)(0)=0=f^(prime)(x)^2=f(5)dot If n is the minimum number of roots of (f^(prime)(x)^2+f^(prime)(x)f^(x)=0 in the interval [0,6], then the value of n/2 is___

Let g: RvecR be a differentiable function satisfying g(x)=g(y)g(x-y)AAx , y in R and g^(prime)(0)=aa n dg^(prime)(3)=bdot Then find the value of g^(prime)(-3)dot

Let f(x) be differentiable for real x such that f^(prime)(x)>0on(-oo,-4), f^(prime)(x) 0on(6,oo), If g(x)=f(10-2x), then the value of g^(prime)(2) is a. 1 b. 2 c. 0 d. 4

Let f(x)a n dg(x) be differentiable functions such that f^(prime)(x)g(x)!=f(x)g^(prime)(x) for any real xdot Show that between any two real solution of f(x)=0, there is at least one real solution of g(x)=0.