Let `f(x)` and `g(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.`

If f(x)=(a x^2+b)^3, then find the function g such that f(g(x))=g(f(x))dot

Let f be a twice differentiable function such that f''(x)=-f(x),a n df^(prime)(x)=g(x),h(x)=[f(x)]^2+[g(x)]^2dot Find h(10)ifh(5)=11

If f(x) and g(x) are non-periodic functions, then h(x)=f(g(x)) is

Let f(x)=x^(2) and g(x)=2x+1 be two real functions. Find (f+g) (x), (f-g) (x), (fg) (x), (f/g) (x) .

Show that the function f defined by f(x)= |1-x + |x|| where x is any real number is continous.

Let f(x+y)=f(x)+f(y)+2x y-1 for all real xa n dy and f(x) be a differentiable function. If f^(prime)(0)=cosalpha, the prove that f(x)>0AAx in Rdot

Let f(x) be a real valued function such that f(0)=1/2 and f(x+y)=f(x)f(a-y)+f(y)f(a-x), forall x,y in R , then for some real a, (a)f(x) is a periodic function (b)f(x) is a constant function (c) f(x)=1/2 (d) f(x)=(cosx)/2

If f: R -> R and g: R ->R are two given functions, then prove that 2min {f(x)-g(x),0}=f(x)-g(x)-|g(x)-f(x)|

f(x)= x + tan x and f is an inverse function of g then g'(x)= ……..

f(x)= x + tan x and f is an inverse function of g then g'(x)= ……..