Home
Class 12
MATHS
Let, f:[0,1]to RR (the set of all real n...

Let, `f:[0,1]to RR` (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)gtf(x),0ltxlt(1)/(4)`

C

`f'(x)ltf(x),0ltxlt(1)/(4)`

D

`f'(x)ltf(x),(3)/(4)ltxlt1`

Text Solution

Verified by Experts

Promotional Banner

Similar Questions

Explore conceptually related problems

Let f:[(1)/(2),1]to RR (the set of all real numbers) be a positive, non-constant and differentiable function such that f'(x)lt2f(x)" and "f((1)/(2))=1 Then the value of int_((1)/(2))^(1)f(x)dx lies in the interval

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) and f(1/2)=1 . Then the value of int f(x)dx lies in the interval for x:[1/2,1] (a) (2e-1,2e) (b) (3-1,2e-1) (c) ((e-1)/2,e-1) (d) (0,(e-1)/2)

If f is a real-valued differentiable function such that f(x) f'(x)lt0 for all real x, then

If f is a real-valued differentiable function such that f(x)f'(x)lt0 for all real x, then -

If f is a function such that f(0)=2, f(1)=3 and f(x+2)=2f(x)-f(x+1) for every real x then f (5) is

Given f(0)=0 and f(x)=(1)/(1-e^(-(1)/(x))) for x ne 0 . Then the function f(x) is -

If f is a real- valued differentiable function such that f(x)f'(x) lt 0 for all real x, then

Let f:(0,oo)->R be a differentiable function such that f'(x)=2-f(x)/x for all x in (0,oo) and f(1) =1 , then

If f'(x)=(1-2sin^2x)/f(x), (f(x) ge 0, AA x in R and f(0)=1) then f(x) is a periodic function with the period

If f:[0,oo) to [0,1), " and " f(x)=(x)/(1+x) then check the nature of the function.