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.