Let `f[0, 1] -> R` (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) leq e^x, x in [0, 1]`.Which of the following is true for `0 lt x lt 1 ?`

Let f:[0,1]rarrR (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) ge e^x, x in [0,1] Which of the following is true for 0 lt x lt 1 ? (A) 0 lt f(x) lt oo (B) -1/2 lt f(x) lt 1/2 (C) -1/4 lt f(x) lt 1 (D) -oo lt f(x) lt 0

Let f:[0,1]rarrR (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) ge e^x, x in [0,1] If the function e^(-x)f(x) assumes its minimum in the interval [0,1] at x=1/4 , which of the following is true? (A) f\'(x) lt f(x), 1/4 lt x lt 3/4 (B) f\'(x) gt f(x), 0 ltxlt1/4 (C) f\'(x) lt f(x), 0 lt x lt 1/4 (D) f\'(x) lt f(x), 3/4 lt x lt 1

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)

If f(0)=f(1)=f(2)=0 and function f (x) is twice differentiable in (0, 2) and continuous in [0, 2]. Then, which of the following is(are) true ?

If f(x) is a twice differentiable function such that f(0)=f(1)=f(2)=0 . Then

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 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 ?