JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let \(f\) be any function continuous on \([\mathrm{a}, \mathrm{b}]\) and twice differentiable on \((a, b) .\) If for all \(x \in(a, b)\) \(f^{\prime}(\mathrm{x})>0\) and \(f^{\prime \prime}(\mathrm{x})<0,\) then for any \(\mathrm{c} \in(\mathrm{a}, \mathrm{b})\) \(\frac{f(\mathrm{c})-f(\mathrm{a})}{f(\mathrm{b})-f(\mathrm{c})}\) is greater than
- A \(\frac{b+a}{b-a}\)
- B \(\frac{b-c}{c-a}\)
- C \(\frac{c-a}{b-c}\)
- D \(1\)
Answer & Solution
Correct Answer
(C) \(\frac{c-a}{b-c}\)
Step-by-step Solution
Detailed explanation
it is clear from graph that \(\mathrm{m}_{1}>\mathrm{m}_{2}\) \(\Rightarrow \quad \frac{f(\mathrm{c})-f(\mathrm{a})}{\mathrm{c}-\mathrm{a}}>\frac{f(\mathrm{b})-f(\mathrm{c})}{\mathrm{b}-\mathrm{c}}\) \(\Rightarrow \quad \frac{f(c)-f(a)}{f(b)-f(c)}>\frac{c-a}{b-c}\)
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