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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 maximum at x=2

B

f has a local minimum at x=2

C

`f''(2) gt f(2)`

D

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

Text Solution

Verified by Experts

The correct Answer is:
B, D
We have
`underset(x to 2)lim (f(x)g(x))/(f'(x)g'(x))=1`
`Rightarrow underset(x to 2)lim (f'(x)g(x)+f(x)g'(x))/(f''(x)g'(x)+f'(x)g''(x))=1" "["Using L' Hospital's rule"]`
`Rightarrow (f'(2)g(2)+f(2)g'(2))/(f''(2)g'(2)+f'(2)g''(2))=1" "[therefore f',f'',g',g' "are continuous"]`
`Rightarrow (f(2)g'(2))/(f''(2)g'(2))=1 " "[therefore f'(2)=g(2)=0]`
`Rightarrow f''(2)=f(2)`
`Rightarrow f''(2) gt 0" "[therefore f:R to (0,oo) therefore f(2) gt 0]`
Thus, we have `f'(2)=0 and f''(2) gt 0`. So, f has a local minimum at x=2
`"Again", f''(2)=f(2)`
`Rightarrow f(2)-f''(2)=0`
`Rightarrow f(x)-f''(x)=0"at "x=2`
`Rightarrow f(x)-f''(x)=0"for at least one"x in R`
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