Let f be any function continuous on [mathrm{a}, mathrm{b}] and twice differentiable on (a, b) . If f

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5. Continuity and Differentiation

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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

$1$

(JEE MAIN-2020)

Solution

it is clear from graph that $\mathrm{m}_{1}>\mathrm{m}_{2}$

$\Rightarrow \quad \frac{f(\mathrm{c})-f(\mathrm{a})}{\mathrm{c}-\mathrm{a}}>\frac{f(\mathrm{b})-f(\mathrm{c})}{\mathrm{b}-\mathrm{c}}$

$\Rightarrow \quad \frac{f(c)-f(a)}{f(b)-f(c)}>\frac{c-a}{b-c}$

Standard 12

Mathematics

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