Exam 3

1. Find the global and local maxima and minima of the following functions. (20%)
   1. $f(x) = 1 + (x+1)^{2}$ for $-2 \le x \le 5$ .
   2. $f(x) = 2 x^{3} + 3 x^{2} + 4$ for $-2 \le x \le 1$ .
2. Let $f(x)=x^{2/3} (6-x)^{1/3}$ ; answer the following. (20%)
   1. Find the intervals of increase and decrease.
   2. Find the local and global extrema.
   3. Find the intervals of concavity and the inflection points.
3. Find values of $b$ and $c$ so that the function $f(x)=x^{2} + b x + c$ has a local minimum at the point $(6, -5)$ . (10%)
4. Find the global maximum and minimum of $x + \sin x$ for $0 \le x \le 2 \pi$ . (10%)
5. Find the point on the parabola $y^{2}=2x$ that is closest to the point $(1,4)$ . (10%)
6. A farmer wants to fence in two rectangular pens as shown illustrated in the following figure. If the farmer has 800 feet of fencing, what dimensions will yield the maximum area for the pens? (10%)
   

7. Suppose that $f(0)=-3$ and $f^{\prime}(x) \le 5$ for all values of $x$ . How large can $f(2)$ possibly be? (10%)
8. Prove that $\ln x \le x-1$ . (10%)