1. Sketch the graph of the function \(f(x) = \frac{x+2}{x^{2}-1},\) and indicate on your graph the intervals where the function is increasing or decreasing, and is concave up or concave down, and any local minimum or maximum values if possible.
2. Sketch the graph of a function that matches the following description: \[ \lim_{x \to -\infty} f(x) = -\infty \qquad f(-3) = -2 \qquad f'(-3) = 0 \qquad f''(-3) = -1 \qquad f(0) = f'(0) = -5 \] \[ f'(1) = 100 \qquad f(2) \gt 0 \qquad f'(2) \lt 0 \qquad f''(2) = 0 \qquad \lim_{x \to \infty} f(x) = 0 \]
3. What are the global minimum and maximum values of the function \(f(x) = x\ln(x)\) on the interval \((0,4]?\)
4. The function \(f(x) = x^{4}-8x^{3}-30x^{2}+12x-17\) has two inflection points. What are their \(x\)-coordinates?
5. Starting with a wire that is 50 inches long, you are to cut the wire into two pieces, bending one piece into a square, and the other into a circle. Find the location where you should cut the wire — i.e. determine the lengths of the two pieces — such that the sum of the areas of the square and circle are minimal.
6. The tops of two utility poles are to be connected by a wire that will also to be anchored to the ground between the poles. The wire ought to be taut. The one pole is 20 ft tall and the other is 10 ft tall, both sticking straight out of the earth. The poles are 30 ft apart, and the ground between the poles is level. Where should the wire be anchored to the ground to minimize the length of wire needed?