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Let f:[0,1]rarrR (the set of all real nu...

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`

A

`0 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`

Text Solution

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The correct Answer is:
D
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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] 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

Knowledge Check

  • Consider the function f:(-oo, oo) -> (-oo ,oo) defined by f(x) =(x^2 - ax + 1)/(x^2+ax+1) ;0 lt a lt 2 . Which of the following is true?

    A
    f(x) is decreasing on (-1,1) and has a local minimum at x=1
    B
    f(x) is increasing on (-1,1) and has a local maximum at x=1
    C
    f(x) is increasing on (-1,1) but has neither a local maximum nor a local minimum at x=1
    D
    f(x) is decreasing on (-1,1) but has neither a local maximum nor a local minimum at x=1

  • Given that f is a real valued differentiable function such that f(x) f'(x) lt 0 for all real x, it follows that

    A
    `f(x)` is an increasing function
    B
    ` f(x)` is a decreasing function
    C
    ` |f(x)|` is an increasing function
    D
    ` |f(x)|` is a decreasing function

  • If f(x)={:{(x", for " 0 le x lt 1),(2", for " x=1),(x+1", for " 1 lt x le 2):} , then f is

    A
    f is continuous at `x=1`
    B
    f is discontinuous at `x=1`
    C
    `underset(x rarr 1^(-))lim f(x)=2`
    D
    `underset(x rarr 1^(+))lim f(x)=1`

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