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et P2 be the 2n sing the idea th or the 2nd Deriu Tote: We are no andwavy argum ense" and "seem Tow write an ac ne first definitic I limit, and the Tote: There are o do it specifica That happens if = f(4) (a), and w Iore specifically n (a) is the sm

et P2 be the 2n sing the idea th or the 2nd Deriu Tote: We are no andwavy argum ense" and "seem Tow write an ac ne first definitic I limit, and the Tote: There are o do it specifica That happens if = f(4) (a), and w Iore specifically n (a) is the sm

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4. Let I be an open interval. Let f be a function defined on I. Let a E I. Assume f is C² on I. Assume f'(a) = 0. As you know, a is a candidate for a local extremum for f. The "2nd Derivative Test" says that, under these circumstances: • IF f"(a) > 0, THEN ƒ has a local minimum at a. • IF f"(a) < 0, THEN ƒ has a local maximum at a. This theorem is easier to justify, and to generalize, using Taylor polynomials.
Transcribed Image Text:4. Let I be an open interval. Let f be a function defined on I. Let a E I. Assume f is C² on I. Assume f'(a) = 0. As you know, a is a candidate for a local extremum for f. The "2nd Derivative Test" says that, under these circumstances: • IF f"(a) > 0, THEN ƒ has a local minimum at a. • IF f"(a) < 0, THEN ƒ has a local maximum at a. This theorem is easier to justify, and to generalize, using Taylor polynomials.
(a) Let P2 be the 2nd Taylor polynomial for f at a. Write an explicit formula for P2. Using the idea that f(x) 2 P2(x) when x is close to a, write an intuitive explanation for the 2nd Derivative Test. Note: We are not asking for a proof yet. Rather, we are asking for a short, simple, handwavy argument that would convince an average student that this result "makes sense" and "seems true". (b) Now write an actual, rigorous proof for the 2nd Derivative Test. You will need to use the first definition of Taylor polynomial (the one in terms of the limit), the definition of limit, and the definition of local extremum. Note: There are many other ways to prove the 2nd Derivative Test, but we want you to do it specifically this way. It will help with the next questions. (c) What happens if f"(a) = 0? In that case, we look at f(3)(a); if it is also 0, then we look at f(4) (a), and we keep looking till we find one derivative that is not 0 at a. More specifically, assume that f is Cn at a for some natural number n > 2 and that f(n) (a) is the smallest derivative that is not 0 at a. In other words, f(k) (a) = 0 for 1<k <n but f(n)(a) # 0. Using the same ideas as in Question 4a, complete the following statements, and give an intuitive explanation for them: • IF THEN f has a local minimum at a. • IF THEN f has a local maximum at a. • IF THEN f does not have a local extremum at a. When you complete this, you will have come up with a new theorem. Let's call it the "Beyond-the-2nd-derivative Test". Make sure your theorem takes care of all possible case
Transcribed Image Text:(a) Let P2 be the 2nd Taylor polynomial for f at a. Write an explicit formula for P2. Using the idea that f(x) 2 P2(x) when x is close to a, write an intuitive explanation for the 2nd Derivative Test. Note: We are not asking for a proof yet. Rather, we are asking for a short, simple, handwavy argument that would convince an average student that this result "makes sense" and "seems true". (b) Now write an actual, rigorous proof for the 2nd Derivative Test. You will need to use the first definition of Taylor polynomial (the one in terms of the limit), the definition of limit, and the definition of local extremum. Note: There are many other ways to prove the 2nd Derivative Test, but we want you to do it specifically this way. It will help with the next questions. (c) What happens if f"(a) = 0? In that case, we look at f(3)(a); if it is also 0, then we look at f(4) (a), and we keep looking till we find one derivative that is not 0 at a. More specifically, assume that f is Cn at a for some natural number n > 2 and that f(n) (a) is the smallest derivative that is not 0 at a. In other words, f(k) (a) = 0 for 1<k <n but f(n)(a) # 0. Using the same ideas as in Question 4a, complete the following statements, and give an intuitive explanation for them: • IF THEN f has a local minimum at a. • IF THEN f has a local maximum at a. • IF THEN f does not have a local extremum at a. When you complete this, you will have come up with a new theorem. Let's call it the "Beyond-the-2nd-derivative Test". Make sure your theorem takes care of all possible case
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