Question

There exist a function f (x), satisfying $f (0) = 1, f'(0) = -1, f (x) > 0$ for all x, and

• A. f ''(x) > 0 for all x
• B. -1 < f " (x) < 0 for all x
• C. - 2 $\le$ f "(x) $\le$ -1 for all
• D. f " (x) < -2 for all x

Answer 1

Answer: (A)

f ''(x) > 0 for all x

Answer 2

$f\left(x\right) =e^{-x}$ is one such function.
Here $f\left(0\right) = 1 ,f'\left(0\right) = - 1, f\left(x\right) > 0, \forall x$
$\therefore f" > 0 \forall x$