AP EAMCET · Maths · Application of Derivatives
If the function \(f(x)=\sqrt{x^2-4}\) satisfies the Lagrange's mean value theorem on \([2,4]\), then the value of C is
- A \(2 \sqrt{3}\)
- B \(-2 \sqrt{3}\)
- C \(\sqrt{6}\)
- D \(-\sqrt{6}\)
Answer & Solution
Correct Answer
(C) \(\sqrt{6}\)
Step-by-step Solution
Detailed explanation
\(f(x)=\sqrt{x^2-4} \Rightarrow f^{\prime}(x)=\frac{x}{\sqrt{x^2-4}}\) \(f^{\prime}(C)=\frac{f(4)-f(2)}{4-2}\) for some \(C \in[2,4]\) \(\frac{C}{\sqrt{C^2-4}}=\frac{2 \sqrt{3}-0}{2}=\sqrt{3}\) \(C^2=3\left(C^2-4\right) \Rightarrow 2 C^2=12 \Rightarrow C^2=\sqrt{6}\)
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