3.4 Higher order derivatives

Only one of the curves could possibly represent $y=f(x)$.

(ii)

Since $f'(x)<0$, we need a decreasing function. This only applies to (ii), (iii), and (v). Since $f''(x)>0$, that means $f'(x)$ is increasing, so the slope of the function must be increasing. In (v), the slope is constant, so $f''(x)=0$–therefore, it's not (v). In (iii), the slope is decreasing, because near $a$ the curve is quite flat ($f'(x)$ near zero) but near $b$ the curve is very steeply decreasing ($f'(x)$ is a large negative number), so (iii) has a negative second derivative. By contrast, in (ii), the line starts out as steeply decreasing ($f'(x)$ is a strongly negative number) and becomes flatter and flatter ($f'(x)$ nears 0), so $f'(x)$ is increasing–in other words, $f''(x)>0$. So, (ii) is the only curve that has $f'(x)<0$ and $f''(x)>0$.