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Let f:Rrarr(0,oo)andg:RrarrR be twice di...

Let `f:Rrarr(0,oo)andg:RrarrR` be twice differentiable functions such that f'' and g'' are continuos functions of R suppose `f'(2)=g(2)=0,f'(2)ne0andg'(2)ne0.` If `underset(xrarr2)lim(f(x)g(x))/(f'(x)g'(x))=1,then`

A

f has a local minimum at x=2

B

f has a local maximum at x=2

C

`f''(2)gtf(x)`

D

`f(X)-f''(x)=0 for at least one x in R

Text Solution

Verified by Experts

The correct Answer is:
1,4
`underset(xrarr2)lim (f(x)g(x))/(f(x)g(x))=1`
`rarr underset(xrarr2)lim(f(x)g(x)+g(x)f(x))/(f(x)g(x)+f(x)g(x))`
`(f(2)g(2)+g(2)f(2))/(f(2)g(2)+f(2)g(2))=1`
`rarr f(2)=f(2)`
Hence option 4 is correct
As `f(2)=f(2) in (0,oo)`
`rarr f(2)gt0`
`rarr` f has local min at x=2
hence option 1 is correct
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