Suppose that for some point x_0. Since f"(x)≥0, it follows that f' is an increasing function, and so f'(x)≥f'(x_0) for all x≥x_0. That means that f(x)→+∞ as x→+∞, contradicting the fact that f(x) is never positive.

Similarly, if f'(x) is ever negative then you can show that f(x)→+∞ as x→–∞.