Suppose `f` is a differentiable real function such that `f(x)+f^(prime)(x)lt=1` for all `x ,` and `f(0)=0,` then the largest possible value of `f(1)` is
(1) `e^(-2)`
(2) `e^(-1)`
(3) `1-e^(-1)`
(4) `1-e^(-2)`

If f: RvecR is a differentiable function such that f^(prime)(x)>2f(x)fora l lxRa n df(0)=1,t h e n : f(x) is decreasing in (0,oo) f^(prime)(x) e^(2x)in(0,oo)

Suppose for a differentiable function f,f(0)=0,f(1)=1 and f(0)=4=f'(1) If g(x)=f(e^(x))*e^(f(x)) then g'(0) is

Let f:[(1)/(2),1]rarr R (the set of all real numbers) be a positive,non-constant,and differentiable function such that f'(x)<2f(x) and f((1)/(2))=1. Then the value of int_((1)/(2))^(1)f(x)dx lies in the interval (a) (2e-1,2e)(b)(3-1,2e-1)(c)((e-1)/(2),e-1)( d) (0,(e-1)/(2))

A function f is such that f(x+1)=f(x)+f(1)+1 for all real values of x . Find f(0) If it is given that f(1)=1, find f(2),f(3) and f(-1) .

Let f: R->R and g: R->R be two non-constant differentiable functions. If f^(prime)(x)=(e^((f(x)-g(x))))g^(prime)(x) for all x in R , and f(1)=g(2)=1 , then which of the following statement(s) is (are) TRUE? f(2) 1-(log)_e2 (c) g(1)>1-(log)_e2 (d) g(1)<1-(log)_e2

If f(x) be a differentiable function for all positive numbers such that f(x.y)=f(x)+f(y) and f(e)=1 then lim_(x rarr0)(f(x+1))/(2x)

Let y=f(x) is a solution of differential equation e^(y)((dy)/(dx)-1)=e^(x) and f(0)=0 then f(1) is equal to