1. The function f(x)=x⁴ - 4x³ + 8x has a critical point at x=1. Use the second derivative test to identify it as a local maximum, a local minimum or neither.

Using calc or computer, graph the following functions. Describe briefly in words the interesting features of the graph including the locations of the critical points and where the function is increasing/decreasing. Then use the derivative and algebra to explain the shape of the graph.

2. Solve the following and show work.

i. f(x)=x³-6x+1

ii. f(x)=3x⁵-5x³

3. How many real roots does the equation x⁵ + x + 7 = 0 have? How do you know?

Use the first derivative to find all critical points and use the second derivative to find all inflections points. Use a graph to identify each critical point as a local maximum, a local minimum or neither.

4. Solve the following, show work.

f(x) = x⁴ - 8x² + 5

5. For f(x) = x³ - 18x² - 10x + 6, find the inflection point algebraically. Graph the function with a calculator or computer and confirm your answer.

6. When I woke up this morning I put on a light jacket, although the temperature was dropping because it seemed the temperature would not go much lower. I was wrong. Around noon a northerly wind blew up and the temp began to drop faster and faster. The worst occurred around 6pm when it started rising again.

a) when was there a critical point in the graph of temperature as a function of time?
b) When was there an inflection point in the graph of temperature as a function of time?

Create a graph of a function on the interval 0 ≤ x ≤ 10 with the following given properties.

7. Has a local minimum at x=3, local maximum x=8 but global maximum and global minimum at the end points of the interval.

8. Has local and global minimum at x=3, local and global minimum at x=8.

9. Plot the graph of f(x)=x³-e^x using a calculator or computer to find all local and global maxima and minima for:

a) 1 ≤ x ≤ 4
b) -3 ≤ x ≤ 2

10. For f(x)=x - ln x, and 0.1 ≤ x ≤ 2, find the values of x for which:

a) f(x) has a local maximum or local minimum. Indicate which ones are maxima and which are minima.
b) F(x) has a global maximum or minimum

11. During a flu outbreak in a school of 763 children, the number of infected children, I, was expressed in the terms of susceptible (but still healthy) children. S, by the expression I = 192ln(S/762)-S+763

What is the maximum possible number of infected children?