(i) Let f : [a, b] → R continuous. Use the Fundamental Theorem of Calculus to show that there isCE (a, b) with4.f(c)(b – a)f(x) dx(ii) Assume that there exists cE (a, b) with| f(x) dx > 0,f (x) dx < 0Show that there exists xo E (a, b) with f(xo)= 0. Hint: Apply the previous result to [a, c] and [c, b]and use the Intermediate Value Theorem.

According to mean value theorem , If f(x)f(x) is continuous over an interval [a,b],[a,b], then there…

4. (i) Let f : [a, b] → R continuous. Use the Fundamental Theorem of Calculus to show that there is CE (a, b) with f(c)(b – a) f(x) dx (ii) Assume that there exists cE (a, b) with 0. f(x) dx > f(x) dx < Show that there exists ro E (a, b) with f(xo): = 0. Hint: Apply the previous result to [a, c] and [c, b] and use the Intermediate Value Theorem.

4. (i) Let f : [a, b] → R continuous. Use the Fundamental Theorem of Calculus to show that there is CE (a, b) with f(c)(b – a) f(x) dx (ii) Assume that there exists cE (a, b) with 0. f(x) dx > f(x) dx < Show that there exists ro E (a, b) with f(xo): = 0. Hint: Apply the previous result to [a, c] and [c, b] and use the Intermediate Value Theorem.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0). The multivariable limit of this function as it approaches the points (x0, y0) and is denoted as:

Question

(i) Let f : [a, b] → R continuous. Use the Fundamental Theorem of Calculus to show that there is
CE (a, b) with
4.
f(c)(b – a)
f(x) dx
(ii) Assume that there exists cE (a, b) with
| f(x) dx > 0,
f (x) dx < 0
Show that there exists xo E (a, b) with f(xo)
= 0. Hint: Apply the previous result to [a, c] and [c, b]
and use the Intermediate Value Theorem.

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