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)``

if f be a differentiable function such that f(x) =x^(2)int_(0)^(x)e^(-t)f(x-t). dt. Then f(x) =

Let f:(0,oo)rarr 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: RvecR is a twice differentiable function such that f"(x)>0fora l lxR ,a n d f(1/2)=1/2,f(1)=1 then: f^(prime)(1)>1 (b) f^(prime)(1)lt=0 (c)1/2 ltf^(prime)(1)lt1 (d)="" 0ltf^(prime)(1)lt="1/2

Let f be a differentiable function such that f(0)=e^(2) and f'(x)=2f(x) for all x in R If h(x)=f(f(x)) ,then h'(0) is equal to

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)

Suppose that f is differentiable for all x and that f^(prime)(x)lt=2fora l lxdot If f(1)=2a n df(4)=8,t h e nf(2) has the value equal to 3 (b) 4 (c) 6 (d) 8

If f is a differentiable function of x, then lim_(h rarr0)([f(x+n)]^(2)-[f(x)]^(2))/(2h)

Prove that the function f(x)=log_(e)x is increasing on (0,oo)

Prove that the function f(x)=(log)_(e)x is increasing on (0,oo)