Q. Let f be a differentiable fuction such that f(f(x)) = x for x=[0,1] suppose f(0) = 1 than integration of (x - f(x))dx from 0 to 1 is ?

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Let f: [0, 1] →R be a function. Suppose the function f is twice differentiable, f (0) =f (1) =0 and satisfies f''(x)-2f'(x) +f(x) ≥e^x, x∈ [0, 1]. Which of the following is true for 0<x<1?

Q. Let f defined on [0, 1] be twice differentiable such that | f"(x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1].

Q. Let f:

[0, 1] \to 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) - 2 f^{'} (x) + f (x) \geq e^{x}, x \in [0, 1]

Which of the following is true for

0 < x < 1

Let

f : [0, 1] \to 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) - 2 f^{'} (x) + f (x) \geq e^{x}, x \in [0, 1]

Which of the following is true for

0 < x < 1

Let $f : [0, 1] \to R$ be a function. Suppose the function $f$ is twice differentiable, $f (0) = f (1) = 0$ and satisfies $f^{''} (x) - 2 f^{'} (x) + f (x) \geq e^{x}$ , $x \in [0, 1]$ . Which of the following is true for $0 < x < 1$ ?

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