Answer
FALSE
Work Step by Step
For $x$:
$f(x)\lt 0$: the function is below x-axis.
$f'(x)\lt 0$: the function is always decreasing.
$f''(x)\gt 0$: function is always concave up.
This is impossible because a function cannot be negative, decreasing and concave up for all $x$ while not having its left end go up above the $x-axis$ toward $\infty$.
Function does not exist.
Hence, the given statement is false.
