Let a < b and let f(x) be a continuously differentiable function on the interval [a, b] with f(x) > 0 for all x € [a, b]. Let A > 0, B > 0 be constants. functional S[y] = [° dæƒ(x)√²+y^², y(a) = A, y(b) = B, a is given by y(x) = A + B [² [ a dw B-A= B 1 √f(w)² - 3²¹ where is a constant satisfying 1 = 1 [dre - √5(23) ² = 3² dw √f(w)² a stationary path of the Using the inequality (which is valid for all real z and u) zu √1 + (z+u)² − √√1 + x² > √1+z²¹ or otherwise, show that the stationary path gives a global minimum of the functional S[y]. ??
Let a < b and let f(x) be a continuously differentiable function on the interval [a, b] with f(x) > 0 for all x € [a, b]. Let A > 0, B > 0 be constants. functional S[y] = [° dæƒ(x)√²+y^², y(a) = A, y(b) = B, a is given by y(x) = A + B [² [ a dw B-A= B 1 √f(w)² - 3²¹ where is a constant satisfying 1 = 1 [dre - √5(23) ² = 3² dw √f(w)² a stationary path of the Using the inequality (which is valid for all real z and u) zu √1 + (z+u)² − √√1 + x² > √1+z²¹ or otherwise, show that the stationary path gives a global minimum of the functional S[y]. ??
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![Let a < b and let f(x) be a continuously differentiable function on the
interval [a, b] with f(x) > 0 for all x € [a, b].
Let A > 0, B > 0 be constants.
functional
S[y] = [° dæƒ(x)√²+y^², y(a) = A, y(b) = B,
a
is given by
y(x) = A + B
x
[²
a
B-A= B
dw
1
√f(w)² - 3²¹
where is a constant satisfying
= 3 [ due
dw
a
stationary path of the
1
√f(w)² – ß²°
Using the inequality (which is valid for all real z and u)
Հա
√1 + (z+u)² − √√√1+z²>
√1+z²
or otherwise, show that the stationary path gives a
global minimum of the functional S[y].
??](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59d44c96-efb1-4f3c-83b3-5a6a84cf94cb%2F23979def-8983-4871-aa23-99353c5f6ff7%2Fekm7bpk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let a < b and let f(x) be a continuously differentiable function on the
interval [a, b] with f(x) > 0 for all x € [a, b].
Let A > 0, B > 0 be constants.
functional
S[y] = [° dæƒ(x)√²+y^², y(a) = A, y(b) = B,
a
is given by
y(x) = A + B
x
[²
a
B-A= B
dw
1
√f(w)² - 3²¹
where is a constant satisfying
= 3 [ due
dw
a
stationary path of the
1
√f(w)² – ß²°
Using the inequality (which is valid for all real z and u)
Հա
√1 + (z+u)² − √√√1+z²>
√1+z²
or otherwise, show that the stationary path gives a
global minimum of the functional S[y].
??
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