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If f(x) is a differentiable real valued ...

If f(x) is a differentiable real valued function satisfying `f''(x)-3f'(x) gt 3 AA x ge 0 and f'(0)=-1,` then `f(x)+x AA x gt 0` is

A

decreasing function of x

B

increasing function of x

C

constant function

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
Given `f''(x)-3f'(x)gt3`
`rArr" "(d)/(dx)(e^(-3x)f'(x))gt 3e^(-3x)`
`rArr" "(d)/(dx)(e^(-3x)f'(x)+e^(-3x))gt0`
`rArr" "e^(-3x)(f'(x)+1)` is increasing funcion
Also `e^(-3x)(f'(x)+1)gt f'(0)+1 AA x gt 0`
`rArr" "f'(x)+1gt0`
`rArr" "f(x)+x` is an increasing function.
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