Suppose that `f(0)=-3` and `f'(x)<=5` for all real values of `x`. Then the largest value of `f(2)` can attain is

Let F:RtoR be a thrice differntiable function. Suppose that F(1)=0,F(3)=-4 and F'(x)lt0 for all x epsilon(1,3) . Let f(x)=xF(x) for all x inR . Then the correct statement(s) is (are)

Consider alpha gt 1 and f:[1/alpha,alpha] rarr [1/alpha,alpha] be bijective function. Suppose that f^(-1)(x)=1/f(x), " for all " in [1/alpha,alpha] . Then f(1) is equal to

If f(x) =x+1/x , prove that : [f(x)]^3 = f(x^3)+3f (1/x) .

Suppose |(f'(x),f(x)),(f''(x),f'(x))|=0 where f(x) is continuous differentiable function with f'(x) !=0 and satisfies f(0)=1 and f'(0)=2 , then f(x)=e^(lambda x)+k , then lambda+k is equal to ..........

Suppose that f(x) is continuous in [0, 1] and f(0) = 0, f(1) = 0 . Prove f(c) = 1 - 2c^(2) for some c in (0, 1)

Let f(x) be a polynomial satisfying f(0)=2 , f'(0)=3 and f''(x)=f(x) then f(4) equals

Suppose f(x) and g(x) are two continuous functions defined for 0<=x<=1 .Given, f(x)=int_0^1 e^(x+1) .f(t) dt and g(x)=int_0^1 e^(x+1) *g(t) dt+x The value of f( 1) equals

If (d/dx)f(x) = 4x^3 - 3/(x^4) such that f(2) = 0 . Then f(x) is

Let f be a differentiable function such that f'(x) = f(x) + int_(0)^(2) f(x) dx and f(0) = (4-e^(2))/(3) . Find f(x) .