Suppose `f` is a real-valued differentiable function defined on `[1,oo]` with `f(1)=1.` Moreover, suppose that `f` satisfies `f^(prime)(x)=1/(x^2+f^2(x))S howt h a tf(x)<1+pi/4AAxgeq1.`

If f(x) is an odd differentiable function defined on (-oo,oo) such that f'(3)=2 , then f'(-3) equal to

IF f(x) is an odd differentiable function defined on (-oo, oo) such that f'(3) = 2, then f'(-3) equal to

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(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 is even function defined on the interval (-5, 5), then real values of x satisfying the equation f(x)=f((x+1)/(x+2)) are

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:(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

Suppose that f(x) is a differentiable function such that f'(x) is continuous, f'(0)=1 and f''(0) does not exist. Let g(x) = xf'(x). Then

Suppose that f(x) is a differentiable function such that f'(x) is continuous, f'(0)=1 and f''(0) does not exist. Let g(x)=xf'(x) . Then-

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 _______