Question
There exist a function $$f\left( x \right),$$ satisfying $$f\left( 0 \right) = 1,\,f'\left( 0 \right) = - 1,\,f\left( x \right) > 0$$ for all $$x,$$ and-
A.
$$f''\left( x \right) > 0$$ for all $$x$$
B.
$$ - 1 < f''\left( x \right) < 0$$ for all $$x$$
C.
$$ - 2 \leqslant f''\left( x \right) \leqslant - 1$$ for all $$x$$
D.
$$f''\left( x \right) < - 2$$ for all $$x$$
Answer :
$$f''\left( x \right) > 0$$ for all $$x$$
Solution :
$$f\left( x \right) = {e^{ - x}}$$ is one such function
$$\eqalign{
& {\text{Here}}\,\,f\left( 0 \right) = 1,\,\,f'\left( 0 \right) = - 1,\,f\left( x \right) > 0,\,\forall \,x \cr
& \therefore \,f''\left( x \right) > 0\,\forall \,x \cr} $$