Is my explanation to this derivative problem correct? Basically, I am trying to prove that number 13 relates to the graph of B, but I was wondering if my explanation was correct. Here is the problem and here is what I wrote.

From the interval of - ∞ to zero, the function is decreasing, so f ' < 0. The interval from zero to + ∞, the function is increasing, so f ' > 0. The graph that corresponds to this is B because the interval from - ∞ to zero, f ' is below zero on the y-axis and the interval from zero to + ∞, f ' is above zero on the y-axis.
Also, for this problem, is there a more efficient way of saying where the function is increasing or decreasing, rather than saying "in this interval, the function is increasing/decreasing," since there are 5 intervals and doesn't give the points on the axis?
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Hint: Your explanation is fine, but you could additionally be more specific.

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*Problem (13): The function $y^\prime$ is linear, so it's a function of the form $y^\prime = kx+d$. Since $(0,0)$ is a point of $y^\prime$ we have $d=0$. We also see the slope $k$ is positive. The function $y$ looks like a parabola and it is going through $0$. Which functions $y, y^\prime$ could be proper candidates?


*Problem (15): We see periodical functions $y$ and $y^\prime$ with $y^\prime(0)=0$. The function $y^\prime$ is $y$ shifted to the left by a quarter of a period, i.e. shifted by $\frac{\pi}{2}$. Which trigonometric functions $y, y^\prime$ could be proper candidates?
