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

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

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

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

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

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

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