Let D°[a, b] = set of all twice differentiable functions on the interval [a, b] with a continuous second derivative. For example, f(x) = sin r is a member of D2[a, b] because it is a twice differentiable function with f"(x) = – sin r and the second derivative f"(x) = – sin r is a continuous function. It can be shown that D²[a, b] is a vector space (you can take it for granted for this question) with the same operations we defined on F([a, b) in the class. Note: Like in F([a, b]), the zero element of D²[a,b] is ƒ(x) = 0 for every r in [a, b], i.e., 0(x) = 0 for every r in [a, b]. (a) If W = {f e D²[0, 1]|f"(x) + 2f'(x) = f(0) + f(1) for very r € [0, 1]}}, show that W is a subspace of D²[0, 1]. (b) Let W = {f e D²[0, 1]| So' f²(x) + f(x)dx = 0}. Is W a subspace of D²[0, 1]?
Let D°[a, b] = set of all twice differentiable functions on the interval [a, b] with a continuous second derivative. For example, f(x) = sin r is a member of D2[a, b] because it is a twice differentiable function with f"(x) = – sin r and the second derivative f"(x) = – sin r is a continuous function. It can be shown that D²[a, b] is a vector space (you can take it for granted for this question) with the same operations we defined on F([a, b) in the class. Note: Like in F([a, b]), the zero element of D²[a,b] is ƒ(x) = 0 for every r in [a, b], i.e., 0(x) = 0 for every r in [a, b]. (a) If W = {f e D²[0, 1]|f"(x) + 2f'(x) = f(0) + f(1) for very r € [0, 1]}}, show that W is a subspace of D²[0, 1]. (b) Let W = {f e D²[0, 1]| So' f²(x) + f(x)dx = 0}. Is W a subspace of D²[0, 1]?
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![Let
D²[a, b] = set of all twice differentiable functions on the interval [a, b] with a continuous second derivative.
For example, f(x) = sin r is a member of D2 [a, b] because it is a twice differentiable function
with f"(x) = - sin z and the second derivative f"(x) = - sin r is a continuous function. It
can be shown that D²[a, b] is a vector space (you can take it for granted for this question)
with the same operations we defined on F([a, b]) in the class. Note: Like in F([a, b]), the
zero element of D²[a,b] is ƒ(x) = 0 for every x in [a, b], i.e., 0(x) = 0 for every a in [a, b].
(a) If W = {f € D²[0, 1]|f"(x) + 2f' (x) = f(0) + f(1) for very x E [0, 1]}}, show that W is
a subspace of D²[0, 1].
(b) Let W = {f e D²[0, 1]| So' f²(x) + f(x)dxr = 0}. Is W a subspace of D²[0, 1]?
%3D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbb1e9de9-a728-40ac-94a8-c6bce5f72bed%2F5fbfd7f3-a9ca-4ed2-bdde-8c5b5800dadc%2Ftl0wu2_processed.png&w=3840&q=75)
Transcribed Image Text:Let
D²[a, b] = set of all twice differentiable functions on the interval [a, b] with a continuous second derivative.
For example, f(x) = sin r is a member of D2 [a, b] because it is a twice differentiable function
with f"(x) = - sin z and the second derivative f"(x) = - sin r is a continuous function. It
can be shown that D²[a, b] is a vector space (you can take it for granted for this question)
with the same operations we defined on F([a, b]) in the class. Note: Like in F([a, b]), the
zero element of D²[a,b] is ƒ(x) = 0 for every x in [a, b], i.e., 0(x) = 0 for every a in [a, b].
(a) If W = {f € D²[0, 1]|f"(x) + 2f' (x) = f(0) + f(1) for very x E [0, 1]}}, show that W is
a subspace of D²[0, 1].
(b) Let W = {f e D²[0, 1]| So' f²(x) + f(x)dxr = 0}. Is W a subspace of D²[0, 1]?
%3D
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