JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let \(S\) be the set of all functions \(f:[0,1] \rightarrow \mathrm{R}\) which are continuous on \([0,1]\) and differentiable on \((0,1) .\) Then for every \(f\) in \(\mathrm{S},\) there exists a \(\mathrm{c} \in(0,1),\) depending on \(f,\) such that
- A \(|f(c)-f(1)|<(1-c)\left|f^{\prime}(c)\right|\)
- B \(|f(c)-f(1)|<\left|f^{\prime}(c)\right|\)
- C \(|f(c)+f(1)|<(1+c)\left|f^{\prime}(c)\right|\)
- D \(\frac{f(1)-f(\mathrm{c})}{1-\mathrm{c}}=f^{\prime}(\mathrm{a})\)
Answer & Solution
Correct Answer
(B) \(|f(c)-f(1)|<\left|f^{\prime}(c)\right|\)
Step-by-step Solution
Detailed explanation
option \((1),(2),(3)\) are incorrect for \(f(x)=\) constant and option ( 4) is incorrect \(\frac{f(1)-f(\mathrm{c})}{1-\mathrm{c}}=f^{\prime}(\mathrm{a})\) where \(\mathrm{c}<\mathrm{a}<1\) (use LMVT) Also for \(f(x)=x^{2}\)
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