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 y=f(x) is an odd differentiable function defined on (-oo,oo) such that f^(prime)(3)=-2,t h e n|f^(prime)(-3)| equals_________.

Let f(x) be a non-constant twice differentiable function defined on (-oo,oo) such that f(x)=f(1-x)a n df^(prime)(1/4)=0. Then (a) f^(prime)(x) vanishes at least twice on [0,1] (b) f^(prime)(1/2)=0 (c) int_(-1/2)^(1/2)f(x+1/2)sinxdx=0 (d) int_(-1/2)^(1/2)f(t)e^(sinpit)dt=int_(1/2)^1f(1-t)e^(sinpit)dt

Let f:(0,oo)toRR be a differentiable function such that f'(x)=2-(f(x))/(x) for all x in(0,oo) and f(1)ne1 . Then-

Let f:(0,oo)->R be a differentiable function such that f'(x)=2-f(x)/x for all x in (0,oo) and f(1) =1 , then

If f(x) is a twice differentiable function such that f(a)=0, f(b)=2, f(c)=-1,f(d)=2, f(e)=0 where a < b < c < de, then the minimum number of zeroes of g(x) = f'(x)^2+f''(x)f(x) in the interval [a, e] is

Let f(x) be a twice-differentiable function and f"(0)=2. The evaluate: lim_(x->0)(2f(x)-3f(2x)+f(4x))/(x^2)

Let f(x) be a twice-differentiable function and f''(0)=2. Then evaluate lim_(xto0) (2f(x)-3f(2x)+f(4x))/(x^(2)).

Let f(x) be differentiable function and g(x) be twice differentiable function. Zeros of f(x),g^(prime)(x) be a,b respectively, (a < b) . Show that there exists at least one root of equation f^(prime)(x)g^(prime)(x)+f(x)g(x)=0 on (a , b) .

Let f(x) be a non-constant twice differentiable function defined on (oo, oo) such that f(x) = f(1-x) and f''(1/4) = 0 . Then (a) f'(x) vanishes at least twice on [0,1] (b) f'(1/2)=0 (c) ∫_{-1/2}^{1/2}f(x+1/2)sinxdx=0 (d) ∫_{0}^{1/2}f(t)e^sinxtdt=∫_{1/2}^{1}f(1−t)e^sinπtdt

Let f: R -> R be a differentiable function such that f(0) = 0,f(pi/2)=3a n df^(prime)(0)=1. If g(x)=int_x^(pi/2)[f^(prime)(t)cos e ct-cottcos e ctf(t)]dt for x(0,pi/2], then (lim)_(x -> 0)g(x)=