Exam (elaborations)
Maxima and Minima solved questions
- Course
- Institution
Maxima and Minima solved questions
[Show more]
Seller
Follow
jureloqoo
Content preview
CHAPTER 13Maxima and Minima
13.1 State the second-derivative test for relative extrema.
I If f ' ( c ) = 0 and /"(e) <0, then f(x) has a relative maximum at c. [See Fig. 13-l(a).] If /'(c) = 0
and /"(c)>0, then f(x) has a relative minimum at c. [See Fig. 13-l(b).] If f ' ( c ) = Q and /"(c) = 0,
we cannot draw any conclusions at all.
(«)
(«)
(*)
(*)
(c)
Fig. 13-1 Fig. 13-2
13.2 State the first-derivative test for relative extrema.
I Assume f ' ( c ) = 0. If/' is negative to the left of c and positive to the right of c—thecase{-,+}—then/has
a relative minimum at c. [See Fig. 13-2(o).] If/' is positive to the left of c and negative to the right of c—the
case{ + , -}—then/has a relative maximum at c. [See Fig. 13-2(6).] If/' has the same sign to the left and to
the right of c—{ + , +} or { —, —}—then/has an inflection point at c. [See Fig. 13-2(c).]
13.3 Find the critical numbers of f(x) = 5 — 2x + x2, and determine whether they yield relative maxima, relative
minima, or inflection points.
I Recall that a critical number is a number c such that /(c) is defined and either /'(c) = 0 or /'(c) does not
exist. Now, f ' ( x ) = -2 + 2x. So, we set -2 + 2x = Q. Hence, the only critical number is x = \. But
/"(AT) = 2. In particular, /"(I) = 2>0. Hence, by the second-derivative test, f(x) has a relative minimum at
*=1.
81
, 82 CHAPTER 13
13.4 Find the critical numbers of /(*) = x l(x - 1) and determine whether they yield relative maxima, relative
minima, or inflection points.
Hence, the critical numbers are x = 0 and x — 2. [x = 1 is not a critical number because /(I) is not
defined.] Now let us compute f"(x).
Hence, /"(0) = — 2 < 0 , and, therefore, by the second-derivative test, f(x) has a relative maximum at
0. Similarly, f"(2) = 2 > 0, and, therefore, /(*) has a relative minimum at 2.
13.5 Find the critical numbers of f(x) = x 3 - 5x 2 - 8x + 3, and determine whether they yield relative maxima,
relative minima, or inflection points.
f /'(*) = 3*2 - 10* - 8 = (3x + 2)(x -4). Hence, the critical numbers are x=4 and x=—\. Now,
f"(x) - 6.v - 10. So, /"(4) = 14 > 0, and, by the second-derivative test, there is a relative minimum at x =
4. Similarly, /"(— f ) = -14, and, therefore, there is a relative maximum at x = - §.
13.6 Find the critical numbers of /(*) = *(* - I)3, and determine whether they yield relative maxima, relative
minima, or inflection points.
f ' ( x ) = x-3(x-l)2 + (x-lY = (x- l) 2 (3x + x-l) = (x- 1) (4x - 1). So, the critical numbers are x = I
and x=\. Now, f"(x) = (x - I)2 • 4 + 2(x - l)(4x - 1) = 2(x - l)[2(jt - 1) + 4* - 1] = 2(jt - 1)(6* - 3)
= 6(jf - l)(2;c - 1). Thus, /"(J) = 6(-i)(-j) = ! >0, and, therefore, by the second-derivative test, there is a
relative minimum at * = j . On the other hand /"(I) = 6 - 0 - 1 =0, and, therefore, the second-derivative
test is inapplicable. Let us use thefirst-derivativetest. f ' ( x ) - (x - 1)2(4* - 1). For j r ^ l , (A:-I) 2 is
positive. Since 4x — 1 has the value 3 when x = l, 4x — 1 > 0 just to the left and to the right of 1.
Hence,/'(*) is positive both on the left and on the right of x = 1, and this means that we have the case { + , +}.
By the first-derivative test, there is an inflection point at x = 1.
13.7 Find the critical numbers of f(x) = sinx — x, and determine whether they yield relative maxima, relative
minima, or inflection points.
I f ' ( x ) = cos x — 1. The critical numbers are the solutions of cos* = 1, and these are the numbers x =
2irn for any integer n. Now, f"(x) = —sinx. So, f"(2irn) = -sin (Iirn) = -0 = 0, and, therefore, the
second-derivative test is inapplicable. Let us use the first-derivative test. Immediately to the left and right
of x = 2irn, c o s j c < l , and, therefore, /'(*) = cos x - 1 < 0. Hence, the case { - , — } holds, and there is
an inflection point at x — 7.-nn.
13.8 Find the critical numbers of f(x) = (x - I) 2 ' 3 and determine whether they yield relative maxima, relative
minima, or inflection points.
I /'(*)= ! ( x - l ) ~ " 3 = §{l/(x-l)" 3 ]. There are no values of x for which /'(*) = 0, but jt = 1 is a
critical number, since/'(l) is not defined. Try the first-derivative test [which is also applicable when/'(c) is not
defined]. To the left of x = \, (x - 1) is negative, and, therefore,/'(.*) is negative. To the right of x = l,
(jt-1) is positive, and, therefore, f(x) is positive. Thus, the case {-,+} holds, and there is a relative
minimum at x = 1.
13.9 Describe a procedure for finding the absolute maximum and absolute minimum values of a continuous function
f(x) on a closed interval [a, b\.
I Find all the critical numbers of f(x) in [a, b]. List all these critical numbers, c,, c 2 , . . ., and add the
endpoints a and b to the list. Calculate /(AC) for each x in the list. The largest value thus obtained is the
maximum value of f(x) on [a, b], and the minimal value thus obtained is the minimal value of f(x) on [a, b].
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller jureloqoo. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $8.61. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
76449 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now