Question

Let `f:R rarr(0,oo) and g:R rarr` R be twice differntiable function such that f' and g' ar continous fucntion on R. Suppose `f(x2)=g(2)=0,f'(2)ne0and g'(2)ne.If lim_(xrarr2) (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

Answer 1

Correct Answer - 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