Science:Math Exam Resources/Courses/MATH102/December 2016/Question A 06/Solution 1

At the points where $x(t)$ has a min/max (slope of tangent line =0), $v(t)=x'(t)$ must be equal to zero, i.e. intersects x-axis. This fact eliminates the option of $x(t)=B(t)$, because if so, we then see that at its maximum point neither $A(t)$ nor $C(t)$ vanishes.

Now we have two choices, for each of which we check whether the graphs match:

• If $x(t)=C(t)$, we see that at $C$'s max and min, $B(t)$ intersects x-axis, this means that $v(t)=x'(t)=B(t)$ so we must have $(x')'(t)=B'(t)=A(t)$ which implies that where $B(t)$ has a max or min $A(t)$ must become zero, however, we've already seen that at $B$'s maximum $A(t)$ is NOT zero. $\Rightarrow {\text{NOT a choice}}$ .

• If $x(t)=A(t)$, we see that at $A$'s max and min, $C(t)$ intersects x-axis, this means that $v(t)=x'(t)=C(t)$ so we must have $(x')'(t)=C'(t)=B(t)$ which implies that where $C(t)$ has a max or min $B(t)$ must become zero, which we see that it is in fact true.

Therefore, the correct choice is $x(t)=A(t)$, $x'(t)=A'(t)=C(t)$, and $x''(t)=\color {blue}A''(t)=C'(t)=B(t)$.

Answer: $\color {blue}(iii)$