MFE 1 - Homework 9
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MFE 1 - Homework 9
The below homework specifications will be enforced. If the specifications are not respected, points
might be deducted, or the homework assignment may not be accepted for grading.
Guidelines for your work
● Write your name (as on the roster) and NetID on the first page.
● If you write on paper, use clean and new sheets of paper and take as much space as necessary.
● Number your pages in the top-right corner, such as 1/3, 2/3, 3/3.
● Use a draft and hand in your final version. Make sure that it
_ is clean and legible;
_ has each problem clearly indicated;
_ does not have anything crossed out or contain notes in the margins;
_ has solutions in which all steps are clearly shown and explained, including all steps of the computations;
_ has grammatically correct complete sentences, including punctuation and spelling; _ is written using correct mathematical terminology and notation;
_ has final answers in exact forms (do not approximate unless otherwise stated).
● You may consult your classmates or other resources (including Campuswire and office hours) for ideas on the problems; however, the solutions you turn in must be in your own words and must reflect your own understanding. Your solutions and write-ups will be checked for textual similarities. You may not copy from, reword, or paraphrase another student’s work or any other resource material; such conduct will be treated as a violation of academic integrity. Remember that you will not learn anything by simply copying, rewording or paraphrasing another person’s work.
Guidelines for Gradescope
● You can either write on blank or lined paper, use a tablet, or type your assignment in LaTeX.
● Your work should be uploaded as a single PDF file (not as separate photos).
● If you write down on paper, scan your work using a scanner or an app such as Camscanner. Make sure that the scans are not blurry and are in portrait mode.
● When you upload this file, match each exercise with the corresponding pages.
Exercise I: Variations (3 × 3 = 9 points)
Study the variations of the chosen functions. In other words, for each function:
● give its domain;
● say on which intervals it is increasing or decreasing.
Show your work and computations. You can summarize your results in a table (but you do no have to).
1. f (x) = (x2 + x − 2)6
2. h(t) = (t + 1)e一t2 /4
3. a(r) = 2r2 ln(r) − r2
Exercise II: Extrema and Concavity (2 × 3 = 6 points)
1. Find the global extrema (minimum and maximum) of the following function on the given interval, and the points where they are attained. Show your computations and your work. You can use a calculator to estimate values if necessary.
h(s) = 
, s ∈ [0, 4]
2. Let f (x) = ln(x2 − 4x + 5). Find the intervals where f is concave up or down.
Exercise III: Why not? (5 × 1 = 5 points)
The following statements regarding a function f defined on an interval [a, b] are false. For each one, give an example (a formula or a graph) to demonstrate that the statement is false, and explain (that is, do the relevant computations to show the statement is false).
1. If f\ (c) = 0 for some c ∈ [a, b], then f has a local extremum at c.
2. If f\\ (c) = 0 for some c ∈ [a, b], then f has an inflection point at c.
3. If f is concave down, then f2 is concave down.
4. There exists c ∈ (a, b) such that f (b) − f (a) = f\ (c).
5. If f (x) > 0 and f\ (x) > 0 for all x, then f is concave up.
2023-04-07