Suppose `f(x)=e^(a x)+e^(b x)` , where `a!=b ,` and that `f^(prime_ prime)(x)-2f^(prime)(x)-15f(x)=0` for all `xdot` Then the value of `(|a b|)/3` is ___

Let inte^x{f(x)-f^(prime)(x)}dx=varphi(x)dot Then inte^xf(x)dx is

Show that inte^x(f(x)+f^(prime)(x))dx=e^xf(x)+C

Let f: R->R be a differentiable function with f(0)=1 and satisfying the equation f(x+y)=f(x)f^(prime)(y)+f^(prime)(x)f(y) for all x ,\ y in R . Then, the value of (log)_e(f(4)) is _______

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___

If f(x)=x|x|, then prove that f^(prime)(x)=2|x|

If f(x)=(e^(x)-e^(-x))/(e^(x)+e^(-x))+2 , then the value of f^(-1)(x) is -

f^(prime)(x)=phi(x) and phi^(prime)(x)=f(x) for all x. Also, f(3)=5 and f^(prime)(3)=4 . Then the value of [f(10)]^2-[phi(10)]^2 is _____

If function f satisfies the relation f(x)*f^(prime)(-x)=f(-x)*f^(prime)(x) for all x ,and f(0)=3, and if f(3)=3, then the value of f(-3) is ______________

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.

Suppose that f(0)=0a n df^(prime)(0)=2, and let g(x)=f(-x+f(f(x)))dot The value of g' (0) is equal to _____