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 ___

Suppose f(x)=e^(a x)+e^(b x),w h e r ea!=b , and that f^(primeprime)(x)-2f^(prime)(x)-15f(x)=0 for all x . Then the product a b is (a) 25 (b) 9 (c) -15 (d) -9

Evaluate: inte^x(f(x)+f^(prime)(x))dx=e^xf(x)+C

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 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 _______

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

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

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 _____

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 _____

Iff(x)=e^(g(x))a n dg(x)=int_2^x(tdt)/(1+t^4), then find the value of f^(prime)(2)

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