Let \( f: R \rightarrow(0, \infty) \) and \( g: R \rightarrow R \) be twice differentiable funct...

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Let \( f: R \rightarrow(0, \infty) \) and \( g: R \rightarrow R \) be twice differentiable functions such that \( f^{\prime \prime} \) and \( g^{\prime \prime} \) are continuous funcitons of \( R \) Suppose \( f^{\prime}(2)=g(2)=0, f^{\prime \prime}(2) \neq 0 \) and \( g^{\prime}(2) \neq 0 \). If \( \lim _{x \rightarrow 2} \frac{f(x) g(x)}{f^{\prime}(x) g^{\prime}(x)}=1 \), then
[More than One Correct option, \( 2016 \mathrm{Adv} \).]
(a) \( f \) has a local minimum at \( x=2 \)
(b) \( f \) has a local maximum at \( x=2 \)
(c) \( f^{\prime \prime}(2)f(2) \)
(d) \( f(x)-f^{\prime \prime}(x)=0 \) for atleast one \( x \in R \)
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