match between a function graph and it's derivative graph

Question

I need help with this question I only manage to match 3 functions:

1-B

2-C

3-D

Answer 1

Observe that curve 4 is increasing on the interval $(-\infty,0)$, decreasing on $(0,+\infty)$, and its function is undefined on $x=0$.

So you want a derivative function that is positive on $(-\infty,0)$, negative on $(0,+\infty)$ and undefined on $x=0$, which corresponds to curve E.

Note that there are other clues that tell you why E is the correct one, like for example: Curve 4 is approaching being flat when $x\to-\infty$ and when $x\to+\infty$, so the curve of the derivative must be approaching $0$ when $x\to-\infty$ and when $x\to+\infty$. Similarly curve 4 is approaching being vertical when $x\to0$ from both sides, so the curve of the derivative must be approaching $\infty$ when $x\to0$ from both sides.

For the curve in 5: It is increasing on $(-\infty,a)$, decreasing on $(a,+\infty)$, and has a stationary point when $x=a$: where $a$ is some positive value just greater than $0$.

However, there is no curve in your pic that has these characteristics: (Positive on $(-\infty,a)$, negative on $(a,+\infty)$, and crosses the x-axis when $x=a$), so the curve 5 has no corresponding derivative curve in these pics.