TS EAMCET · Maths · Application of Derivatives
Consider all functions given in List-I in the interval \([1,3]\). The List- 2 has the values of ' \(c\) ' obtained by applying Lagrange's mean value theorem on the functions of List-1. Match the functions and values of 'c'.
| List - 1 | List - 2 | ||
|---|---|---|---|
| A | \(|x-1|\) | I | \(2 \log \left(e^3+e^2\right)\) |
| B | log x | II | 2 |
| C | \(x^2+x+1\) | III | \(\log _3 e^2\) |
| D | \(e^x\) | IV | \(\sqrt{2}\) |
| V | \(\log \left(\frac{e^3-e}{2}\right)\) | ||
- A \(\mathrm{A}-\mathrm{II}, \mathrm{B}-\mathrm{V}, \mathrm{C}-\mathrm{IV}, \mathrm{D}-\mathrm{III}\)
- B \(\mathrm{A}-\mathrm{II}, \mathrm{B}-\mathrm{I}, \mathrm{C}-\mathrm{IV}, \mathrm{D}-\mathrm{III}\)
- C \(\mathrm{A}-\mathrm{IV}, \mathrm{B}-\mathrm{V}, \mathrm{C}-\mathrm{II}, \mathrm{D}-\mathrm{I}\)
- D \(\mathrm{A}-\mathrm{IV}, \mathrm{B}-\mathrm{III}, \mathrm{C}-\mathrm{II}, \mathrm{D}-\mathrm{V}\)
Answer & Solution
Correct Answer
(D) \(\mathrm{A}-\mathrm{IV}, \mathrm{B}-\mathrm{III}, \mathrm{C}-\mathrm{II}, \mathrm{D}-\mathrm{V}\)
Step-by-step Solution
Detailed explanation
A: \(f(x) = |x-1|\) \(f(x) = x-1\) for \(x \in [1,3]\) \(f'(x) = 1\) \(\frac{f(3)-f(1)}{3-1} = \frac{2-0}{2} = 1\) \(f'(c) = 1 \Rightarrow 1 = 1\) \(c = \sqrt{2}\) is in \((1,3)\). (Match IV) B: \(f(x) = \log x\) \(f'(x) = \frac{1}{x}\)…
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