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 differentiable functions in R a n d f(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 fa n dg be differentiable on [0,1] such that f(0)=2,g(0)=0,f(1)=6a n dg(1)=2. Show that there exists c in (0,1) such that f^(prime)(c)=2g^(prime)(c)dot

Let f(x)a n dg(x) be differentiable for 0lt=xlt=2 such that f(0)=2,g(0)=1,a n df(2)=8. Let there exist a real number c in [0,2] such that f^(prime)(c)=3g^(prime)(c)dot Then find the value of g(2)dot

Let f(x)a n dg(x) be differentiable for 0lt=xlt=2 such that f(0)=2,g(0)=1,a n df(2)=8. Let there exist a real number c in [0,2] such that f^(prime)(c)=3g^(prime)(c)dot Then find the value of g(2)dot

If f(x)a n dg(x) are differentiable functions for 0lt=xlt=1 such that f(0)=10 ,g(0)=2,f(1)=2,g(1)=4, then in the interval (0,1)dot (a) f^(prime)(x)=0 for a l lx (b) f^(prime)(x)+4g^(prime)(x)=0 for at least one x (c) f(x)=2g'(x) for at most one x (d) none of these

If f(x)a n dg(x) are differentiable functions for 0lt=xlt=1 such that f(0)=10 ,g(0)=2,f(1)=2,g(1)=4, then in the interval (0,1)dot (a) f^(prime)(x)=0 for a l lx (b) f^(prime)(x)+4g^(prime)(x)=0 for at least one x (c) f(x)=2g'(x) for at most one x (d) none of these

Let f(x)a n dg(x) be differentiable function in (a , b), continuous at aa n db ,a n dg(x)!=0 in [a , b]dot Then prove that (g(a)f(b)-f(a)g(b))/(g(c)f^(prime)(c)-f(c)g^(prime)(c))=((b-a)g(a)g(b))/((g(c))^2)

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.

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 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