AP EAMCET · Maths · Differentiation
Observe the following statements
I. \(f(x)=a x^{41}+b x^{-40} \Rightarrow \frac{f^{\prime \prime}(x)}{f(x)}=1640 x^{-2}\)
II. \(\frac{d}{d x} \tan ^{-1}\left(\frac{2 x}{1-x^2}\right)=\frac{1}{1+x^2}\)
Which of the following is correct?
- A \(\mathrm{I}\) is true, but \(\mathrm{II}\) is false
- B Both I and II are true
- C Neither I nor II is true
- D I is false, but II is true
Answer & Solution
Correct Answer
(A) \(\mathrm{I}\) is true, but \(\mathrm{II}\) is false
Step-by-step Solution
Detailed explanation
I. \(\begin{aligned} & f(x)=a x^{41}+b x^{-40} \\ & f^{\prime}(x)=41 a x^{40}-40 b x^{-41} \\ & f^{\prime \prime}(x)=1640 a x^{39}+1640 b x^{-42} \end{aligned}\) Now, \(\frac{f^{\prime \prime}(x)}{f(x)}=\frac{1640\left(a x^{39}+b x^{-42}\right)}{a x^{41}+b x^{-40}}=1640 x^{-2}\)…
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