JEE Mains · Maths · STD 12 - 6. Application of derivatives
For the function \(\mathrm{f}(\mathrm{x})=(\cos \mathrm{x})-\mathrm{x}+1, \mathrm{x} \in \mathbb{R}\), between the following two statements (\(S1\)) \(f(x)=0\) for only one value of \(x\) is \([0, \pi]\). (\(S2\)) \(\mathrm{f}(\mathrm{x})\) is decreasing in \(\left[0, \frac{\pi}{2}\right]\) and increasing in \(\left[\frac{\pi}{2}, \pi\right] .\)
- A Both (\(S1\)) and (\(S2\)) are correct
- B Only (\(S1\)) is correct
- C Both (\(S1\)) and (\(S2\)) are incorrect
- D Only (\(S2\)) is correct
Answer & Solution
Correct Answer
(B) Only (\(S1\)) is correct
Step-by-step Solution
Detailed explanation
\( f(x)=\cos x-x+1 \) \( f(x)=-\sin x-1\) \(\mathrm{f}\) is decreasing \(\forall \mathrm{x} \in \mathrm{R}\) \( f(x)=0 \) \( f(0)=2, f(\pi)=-\pi\) \(\mathrm{f}\) is strictly decreasing in \([0, \pi]\) and \(\mathrm{f}(0) . \mathrm{f}(\pi)<0\) \(\Rightarrow\) only one solution of…
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