The figure shows $y=f^{\prime }(x)$ on $[-4,4]$. On what interval is $f$ decreasing?

The figure shows $y=f^{\prime }(x)$ on $[-3,3]$. Where is the graph of $f$ concave up?

The figure shows $y=f^{\prime }(x)$ on $[-3,4]$. At what value of $x$ does $f$ have a relative maximum?

The figure shows $y=f^{\prime }(x)$ on $[-3,3]$. On what interval is $f$ concave down?

The figure shows $y=f^{\prime }(x)$ on $[-3,3]$. At what value of $x$ does $f$ have a relative minimum?

The figure shows $y=f^{\prime }(x)$ on $[0,2\pi ]$. On what interval is $f$ increasing?

The figure shows $y=f^{\prime }(x)$ on $[0,2\pi ]$. Where is the graph of $f$ concave up?

A differentiable function has continuous $f^{\prime }$, whose only zero on $[1,3]$ is $x=2$. Given $f^{\prime }(1)=-2$ and $f^{\prime }(3)=4$, what follows?

A twice-differentiable function has $f^{\prime \prime }(-1)=3$, $f^{\prime \prime }(0)=0$, and $f^{\prime \prime }(1)=-2$. What conclusion is supported?

A function is continuous on $[0,2]$ and differentiable on $(0,2)$. The table gives $f(0)=1$ and $f(2)=7$. What must be true?

A differentiable function has continuous $f^{\prime }$ and no zeros except $x=2$. The table gives $f^{\prime }(0)=-5$, $f^{\prime }(1)=-1$, $f^{\prime }(2)=0$, and $f^{\prime }(3)=4$. On what interval is $f$ decreasing?

A twice-differentiable function has continuous $f^{\prime \prime }$, whose only zero on $[1,5]$ is $x=3$. The table gives $f^{\prime \prime }(1)=-2$ and $f^{\prime \prime }(5)=4$. What follows?

On $[0,4]$, the only critical points of $f$ are $x=1$ and $x=3$. The table gives $f(0)=2$, $f(1)=7$, $f(3)=-1$, and $f(4)=3$. Where is the absolute maximum?

A differentiable function has $f^{\prime }(x)>0$ just left of $x=2$ and $f^{\prime }(x)<0$ just right of $x=2$. What justification establishes that $f$ has a relative maximum at $x=2$?

A differentiable function satisfies $f^{\prime }(x)>0$ for every $x\in (1,5)$. What justification establishes that $f$ is increasing on $[1,5]$?

A twice-differentiable function has $f^{\prime \prime }(x)<0$ just left of $x=0$ and $f^{\prime \prime }(x)>0$ just right of $x=0$. What justification establishes an inflection point at $x=0$?

A function is continuous on $[0,6]$ and differentiable on $(0,6)$. Its only critical points are $x=2$ and $x=5$. What justification establishes an absolute maximum on the interval?

A function $f$ is continuous on $[0,4]$, differentiable on $(0,4)$, and satisfies $f(0)=f(4)$. What justification establishes that some $c\in (0,4)$ has $f^{\prime }(c)=0$?

A function $g$ is continuous on $[1,3]$, differentiable on $(1,3)$, and satisfies $g(1)=g(3)$. What conclusion is justified?

A twice-differentiable function satisfies $f^{\prime }(5)=0$ and $f^{\prime \prime }(5)>0$. What justification establishes a relative minimum at $x=5$?

The temperature $T(t)$ in a greenhouse is measured in degrees Celsius. At $t=3$ hours, $T^{\prime }(3)=0$ and $T^{\prime \prime }(3)<0$. What does this mean?

A robot's position $s(t)$ is measured in meters. At $t=4$ seconds, $s^{\prime }(t)$ changes from positive to negative. What must occur at 4 seconds?

The concentration $C(t)$ of a solution is measured in milligrams per liter. If $C^{\prime }(2)>0$ and $C^{\prime \prime }(2)<0$, how is the concentration changing at 2 minutes?

The water volume $V(t)$ in a tank is measured in liters. If $V^{\prime }(6)<0$ and $V^{\prime \prime }(6)>0$, what describes the volume at 6 minutes?

The height $h(t)$ of a drone is measured in meters. At $t=8$ seconds, $h^{\prime }(8)=0$ and $h^{\prime \prime }(8)>0$. What does this mean?

A differentiable function has continuous $f^{\prime }$, whose only zero on $[0,4]$ is $x=2$. Given $f^{\prime }(0)<0$ and $f^{\prime }(4)>0$, what must be true?

A differentiable function has continuous $f^{\prime }$ on $[1,3]$. If $f^{\prime }(1)<0$ and $f^{\prime }(3)>0$, what must be true?

A function $f$ is continuous on $[0,5]$. What must be true?

A differentiable function satisfies $f^{\prime }(x)<0$ for every $x\in (0,6)$. What must be true?

A twice-differentiable function has $f^{\prime \prime }(x)<0$ just left of $x=3$ and $f^{\prime \prime }(x)>0$ just right of $x=3$. What must be true?