AP Calculus AB Unit 5: Analytical Applications of Differentiation

If $f^{\prime \prime }(x)<0$ on an interval, then the graph of $f$ is:

A particle moves so that $x(t)=t^3-6t^2+9t$ for $t\ge 0$. On which interval(s) is the particle's speed increasing?

Using the Second Derivative Test, classify the critical point $x=1$ of $f(x)=x^3-3x+2$.

The graph of $f^{\prime }$ is shown (not provided here — described): $f^{\prime }(x)>0$ for $x<2$, $f^{\prime }(2)=0$, and $f^{\prime }(x)<0$ for $x>2$. What can be concluded about $f$ at $x=2$?

Find all inflection points of $f(x)=x^4-4x^3$.

Find the absolute maximum of $f(x)=-x^2+4x+1$ on the interval $[0,5]$.

The function $f$ is continuous on $[2,8]$, differentiable on $(2,8)$, with $f(2)=1$ and $f(8)=13$. Which theorem guarantees a $c\in (2,8)$ where $f^{\prime }(c)=2$?

A function $f$ is continuous on $[-3,5]$. Which theorem guarantees that $f$ attains both a maximum and a minimum value on this interval?

A farmer wants to enclose a rectangular plot next to a river (no fence needed along the river) using $200$ m of fencing. What dimensions maximize the enclosed area?

A function $f$ has $f^{\prime }(c)=0$ and $f^{\prime \prime }(c)=0$. What can be concluded?

Let $f(x)=x^3-3x^2-9x+5$. On what interval(s) is $f$ increasing?

If $f^{\prime }(x)>0$ on $(a,b)$, then $f$ is: