Let `f:[1/2,1]->R` (the set of all real numbers) be a positive, non-constant, and differentiable function such that `f^(prime)(x)<2f(x)a n df(1/2)=1` . Then the value of `int_(1/2)^1f(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)`

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))

Let f:[0,1]toR (the set of all real numbers ) be a function. Suppose the function f is twice differentiable,f(0)=f(1)=0 and satisfies f''(x)-2f'(x)+f(x)gee^(2),x in [0,1] Consider the statements. I. There exists some x in R such that, f(x)+2x=2(1+x^(2)) (II) There exists some x in R such that, 2f(x)+1=2x(1+x)

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

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

Let R be the set of real numbers and let f:RtoR be a function such that f(x)=(x^(2))/(1+x^(2)) . What is the range of f?

Let R be the set of all real numbers and let f be a function from R to R such that f(X)+(X+1/2)f(l-X)=1, for all xinR." Then " 2f(0)+3f(1) is equal to-

If R denotes the set of all real numbers, then the function f : R to R defined by f (x) =[x] is

Let b be a nonzero real number , Suppose f : R to R is differentiable function such that f(0) = 1 . If the derivative f' of satisfies the equation f'(x) = (f(x))/(b^(2)+x^(2)) for all x in R R ' , then which of the following statements is/are True ?

If f:R rarr R is a differentiable function such that f(x)>2f(x) for all x in R and f(0)=1, then