Question

If '' > 0$for all$

• A. $' (1) \leq 0$
• B. $0 < ' 1 = \frac{1}{2}$
• C. $\frac{1}{2} < ' 1 = 1$
• D. $' (1) > 1$

Answer 1

Answer: (D)

$' (1) > 1$

Answer 2

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
$\therefore f (1)> f '(\alpha)$
$f '(1)>1$