# WRITTEN ASSIGNMENT 8

```WRITTEN ASSIGNMENT 8
Hand in full solutions to the questions below. Make sure you justify all your work and include complete
arguments and explanations. Your asnwers must be clear and neatly written, as well as legible (no tiny
at the top of each page.
1. Consider the graph showed in question 1 in Assignment 2.6 on WeBWorK. This is graph of the first
derivative f 0 of a function f . Reproduce the graph on your paper. Then determine the following
information:
(a) intervals where the f is concave up and concave down.
(b) the x-ccordinates of any inflection points of f , if they exist.
(c) sketch the graph of the second derivative f 00 .
(d) sketch the graph of the function f .
2. Sketch the graph of a continuous function f that satisfies all of the given conditions:
• f 0 (0) = f 0 (2) = f 0 (4) = 0
• f 0 (x) &gt; 0 if x &lt; 0 or 2 &lt; x &lt; 4
• f 0 (x) &lt; 0 if 0 &lt; x &lt; 2 or x &gt; 4
• f 00 (x) &gt; 0 if 1 &lt; x &lt; 3, f 00 (x) &lt; 0 if x &lt; 1 or x &gt; 3
Make sure you label the coordinates of all local extreme values and inflection points on your axes.
3. Let T (t) be the temperature at time t in Vancouver. Suppose that at time t = 3 you feel uncomfortably
hot. The data below correspond to four possible scenarios. In each case, explain whether you would
expect your level of comfort (relative to air temperature) to improve or get worse. Explain why.
(a) T 0 (3) = 2, T 00 (3) = 4
(b) T 0 (3) = 2, T 00 (3) = −4
(c) T 0 (3) = −2, T 00 (3) = 4
(d) T 0 (3) = −2, T 00 (3) = −4
4. If f is continuous, positive, and concave upward on an interval [a, b], what is the concavity of the
function g(x) = [f (x)]2 on the interval [a, b]? Justify your claim.
5. Explain why the function g(x) = x|x| has and inflection point at (0, 0) even though g 00 (0) does not