Let X, Y be Banach spaces and U be an open subset of X. +1)-times continuously differentiable mapping of U into ent joining a and a + h is in U, we have 1 f(a+h) = f(a) + f'(a)h + ¹ ƒ" (a) h (²) + ... + ¹ f(n) (a)
Let X, Y be Banach spaces and U be an open subset of X. +1)-times continuously differentiable mapping of U into ent joining a and a + h is in U, we have 1 f(a+h) = f(a) + f'(a)h + ¹ ƒ" (a) h (²) + ... + ¹ f(n) (a)
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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![Let X, Y be Banach spaces and U be an open subset of X. Letf: U→Y
be (n + 1)-times continuously differentiable mapping of U into Y. Then, if the
segment joining a and a + h is in U, we have
1
1_
f(a+h) = f(a) + f'(a)h + — ƒ" (a)h(²) + ... + ¹ f(n) (a) h(n)
[2f"
n
Here h(n)= (h.h...h), n-times.
+
1
·(1-t) f(n+¹)(a + th) h(n+1)
n
dt.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87c0e676-9e12-4bbb-b553-711f5739442d%2F0b020e13-f8bb-4aee-b344-8951735239ad%2F4o1krpj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let X, Y be Banach spaces and U be an open subset of X. Letf: U→Y
be (n + 1)-times continuously differentiable mapping of U into Y. Then, if the
segment joining a and a + h is in U, we have
1
1_
f(a+h) = f(a) + f'(a)h + — ƒ" (a)h(²) + ... + ¹ f(n) (a) h(n)
[2f"
n
Here h(n)= (h.h...h), n-times.
+
1
·(1-t) f(n+¹)(a + th) h(n+1)
n
dt.
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