(a) If a1 < · ·:< an, find the minimum value of f(x) = ) (x – a;)². *(b) Now find the minimum value of f(x) = } ]x a;|. This is a prob- lem where calculus won't help at all: on the intervals between the a's the functionf is linear, so that the minimum clearly occurs at one of the a;, and these are precisely the points where f is not differ- entiable. However, the answer is easy to find if you consider how f(x) changes as you pass from one such interval to another. *(c) Let a > 0. Show that the maximum value of 1 + 1+ |x|'1+ x – a f(x) %3| is (2 + a)/(1 + a). (The derivative can be found on each of the intervals (- 0, 0), (0, a), and (a, ∞) separately.)
(a) If a1 < · ·:< an, find the minimum value of f(x) = ) (x – a;)². *(b) Now find the minimum value of f(x) = } ]x a;|. This is a prob- lem where calculus won't help at all: on the intervals between the a's the functionf is linear, so that the minimum clearly occurs at one of the a;, and these are precisely the points where f is not differ- entiable. However, the answer is easy to find if you consider how f(x) changes as you pass from one such interval to another. *(c) Let a > 0. Show that the maximum value of 1 + 1+ |x|'1+ x – a f(x) %3| is (2 + a)/(1 + a). (The derivative can be found on each of the intervals (- 0, 0), (0, a), and (a, ∞) separately.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Solve question 4 completely
![8:37 AM Ø
VOLTE 4 28%
3
= x +
x?
(ii) f(x)
x?
(iii) f(x)
х2 — 1
1
(iv) f(x)
1 + x?
4.
(a) If a1 < · ·. = > (x – a;)?.
< an, find the minimum value of f(x)
*(b) Now find the minimum value of f(x) = ) ]* -
a;l. This is a prob-
lem where calculus won't help at all: on the intervals between the
a;'s the function f is linear, so that the minimum clearly occurs at one
of the a;, and these are precisely the points where f is not differ-
entiable. However, the answer is easy to find if you consider how
f(x) changes as you pass from one such interval to another.
*(c) Let a > 0. Show that the maximum value of
1
1
+
1 + |x|
f(x) =
1+ |x - al
is (2 + a)/(1 + a). (The derivative can be found on each of the
intervals (- o, 0), (0, a), and (a, o) separately.)
For each of the following functions, find all local maximum and mini-
mum points.
5.
x, x * 3, 5, 7, 9
5,
X = 3
-3, x = 5
9, x = 7
7, х 3D 9.
(i) f(x)
=
0,
1/g, x = p/g in lowest terms.
x rational
| 0, x irrational.
x irrational
(ii) f(x)
х,
(iii) f(x)
=
* = 1/n for some n in N
0, otherwise.
(iv) f(x)
1, if the decimal expansion of x contains a 5
0, otherwise.
(v) f(x)
6. (a) Let (x0, yo) be a point of the plane, and let L be the graph of the
function f(x) = mx + b. Find the point F such that the distance
from (xo, yo) to (x, f(x)) is smallest. [Notice that minimizing this
distance is the same as minimizing its square. This may simplify
the computations somewhat.]
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Transcribed Image Text:8:37 AM Ø
VOLTE 4 28%
3
= x +
x?
(ii) f(x)
x?
(iii) f(x)
х2 — 1
1
(iv) f(x)
1 + x?
4.
(a) If a1 < · ·. = > (x – a;)?.
< an, find the minimum value of f(x)
*(b) Now find the minimum value of f(x) = ) ]* -
a;l. This is a prob-
lem where calculus won't help at all: on the intervals between the
a;'s the function f is linear, so that the minimum clearly occurs at one
of the a;, and these are precisely the points where f is not differ-
entiable. However, the answer is easy to find if you consider how
f(x) changes as you pass from one such interval to another.
*(c) Let a > 0. Show that the maximum value of
1
1
+
1 + |x|
f(x) =
1+ |x - al
is (2 + a)/(1 + a). (The derivative can be found on each of the
intervals (- o, 0), (0, a), and (a, o) separately.)
For each of the following functions, find all local maximum and mini-
mum points.
5.
x, x * 3, 5, 7, 9
5,
X = 3
-3, x = 5
9, x = 7
7, х 3D 9.
(i) f(x)
=
0,
1/g, x = p/g in lowest terms.
x rational
| 0, x irrational.
x irrational
(ii) f(x)
х,
(iii) f(x)
=
* = 1/n for some n in N
0, otherwise.
(iv) f(x)
1, if the decimal expansion of x contains a 5
0, otherwise.
(v) f(x)
6. (a) Let (x0, yo) be a point of the plane, and let L be the graph of the
function f(x) = mx + b. Find the point F such that the distance
from (xo, yo) to (x, f(x)) is smallest. [Notice that minimizing this
distance is the same as minimizing its square. This may simplify
the computations somewhat.]
Alee fo
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