Let f be any function continuous on [a, b] and twice differentiab… - Vidyadip

MCQ

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
$\frac{b-c}{c-a}$
✓
$\frac{c-a}{b-c}$
D
$1$

✓

Answer

Correct option: C.

$\frac{c-a}{b-c}$

c
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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