AP Calculus BC Unit 5: Analytical Applications of Differentiation

For $f(x)=x^3-6x^2+9x$, on what intervals is $f$ increasing?

Let $f$ be twice differentiable with $f^{\prime }(2)=0$, $f^{\prime \prime }(2)=0$, and $f^{\prime \prime \prime }(2)\ne 0$. What can be concluded about $x=2$?

A closed cylinder must hold volume $V=250\pi$ cm${}^3$. Find the radius $r$ that minimizes total surface area $S=2\pi r^2+2\pi rh$.

A farmer has $200$ m of fencing to enclose a rectangular field against a straight river (no fence on the river side). What dimensions maximize the enclosed area?

On what interval is $f(x)=x{e}^{-x}$ concave down?

A function has $f^{\prime }(x)=(x-1{)}^2(x-3)$. On which interval is $f$ strictly decreasing?

For $f(x)=x^2$ on $[1,3]$, find the value $c$ guaranteed by the Mean Value Theorem.

For $g(x)=x^4-8x^2$, use the Second Derivative Test to classify $x=0$ and $x=\pm 2$.

Let $f(x)=x^3+a{x}^2+bx$ have a local maximum at $x=1$ and a local minimum at $x=3$. Find $a$ and $b$.

Using the MVT on $f(x)=\sqrt{x}$ over $[4,9]$, find $c$ such that $f^{\prime }(c)$ equals the average rate of change.

Find all inflection points of $h(x)=x^4-6x^2+5$.

Let $f(x)=x^3-3x$ on $[-2,2]$. By the Extreme Value Theorem, the absolute maximum value of $f$ on this interval is: