- Find the absolute extrema if they exist, as well as all values of
\(x\) where they occur.
- \(f(x) = x^3 - 2x^2 - 4x + 8\) on \([-1,0]\).
- \(h(x) = xe^{-kx}\) on \([0,\infty)\), for a constant \(k > 0\).
- \(k(x) = \sin(x)\) on \((-\infty,\infty)\).
- \(\ell(x) = \begin{cases} x^2 &\quad x \geq 0 \\ 0 &\quad x < 0
\end{cases}\) on \((-\infty, \infty)\).
- \(m(x) = x\sin(x)\) on \([0, \infty)\).
- A hunter is at a point along the river bank. He wants to get to his
cabin, located 3 miles north and 8 miles west. He can travel west
along the river at the speed of 5 mph, but only travel 2 mph
through the wilderness between the river and his cabin. How far
upriver should he travel in order to reach the cabin in minimum
time?
- Determine whether the demand is elastic or not at the indicated
values: \(q = 400 - 0.2 p^2\), for \(p = \$20\) and \(p = \$40\).
- Suppose we want to cut a rectangular beam from a cylindrical log of
radius \(10\) inches.
- Show that the beam of a maximal cross sectional area is a
square.
- Suppose the strength of a beam is proportional to the product of
its width and the square of its depth (width and depth are two
sides of the cross section rectangle). Find the dimension of the
strongest beam that can be cut from the log.