Question:
If '' > 0$ for all $
If '' > 0$ for all $
Updated On: Jun 14, 2022
- $' (1) \leq 0$
- $0 < ' 1 = \frac{1}{2}$
- $\frac{1}{2} < ' 1 = 1 $
- $' (1) > 1$
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The Correct Option is D
Solution and Explanation
Let $h ( x )= f ( x )- x$
$h \left(\frac{1}{2}\right)=0= h (1)$
$\Rightarrow h '(\alpha)=0$ for some $\alpha \in(0,1)$ by rolle's theorem
$f'(\alpha)=1$
as $f'(x)>0 $
$\Rightarrow f(x)$ is increasing
$\therefore f (1)> f '(\alpha)$
$f '(1)>1$
$h \left(\frac{1}{2}\right)=0= h (1)$
$\Rightarrow h '(\alpha)=0$ for some $\alpha \in(0,1)$ by rolle's theorem
$f'(\alpha)=1$
as $f'(x)>0 $
$\Rightarrow f(x)$ is increasing
$\therefore f (1)> f '(\alpha)$
$f '(1)>1$
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