I'm confused on why there is a maximum at R. If I flipped the acceleration graph it looks like a continuously increasing function with no max or min to me. Could someone help me understand this? 8a (linear graph) is $f''(x)$
2026-04-04 04:55:46.1775278546
Finding the max with velocity and acceleration graph
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The first derivative of $f(x)$ is zero at $x=r$, so $r$ is clearly a critical point. We see that $f'(x)$ is positive to the left of $r$ and negative to the right. With all these facts, the "first derivative test" shows that $f(x)$ is a maximum at $x=r$.
The First Derivative Test can be proven by using the Mean Value Theorem on any closed interval $[s,r]$ for $s$ slightly to the left of $r$ and again on $[r,t]$ for $t$ slightly to the right of $r$.
Note that only the first diagram was used here.