et [a, b] be a real interva b]→X is continuous, the is continuously differentia

College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
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Q.10. (a) Let [a, b] be a real interval and X be a
Banach space
(i) If f:[a,b] → X is continuous, then prove that
F(t)=ff(s)ds is continuously differentiable in (a, b)
a
and F = f.
(ii) If f is continuously differentiable in an open
interval containing [a, b], then prove that
f(b)-f(a)=f(t)dt
Transcribed Image Text:Q.10. (a) Let [a, b] be a real interval and X be a Banach space (i) If f:[a,b] → X is continuous, then prove that F(t)=ff(s)ds is continuously differentiable in (a, b) a and F = f. (ii) If f is continuously differentiable in an open interval containing [a, b], then prove that f(b)-f(a)=f(t)dt
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