Let f : R → (0, ∞) and g : R → R be twice differentiable functions such that f ′′ and g…
Mathematics · JEE Advanced · NTA Exams — Limit, Continuity and Differentiability
Let f : R → (0, ∞) and g : R → R be twice differentiable functions such that f ′′ and g′′ are continuous functions on R. Suppose f ′(2) = g(2) = 0, f ′′(2) ≠ 0 and g′(2) ≠ 0. If \(\lim _{x \rightarrow 2} \frac{f(x) g(x)}{f^{\prime}(x) g^{\prime}(x)}=1,\) then
- f has a local minimum at x = 2
- f has a local maximum at x = 2
- f ′′(2) > f (2)
- f (x) – f ′′(x) = 0 for at least one x ∈ R
Answer
(D) f ( x ) – f ′′ ( x ) = 0 for at least one x ∈ R
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