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16.J. Show that every polynomial of odd degree and real coefficients has a real root. Show that the polynomial p(x) = x¢ + 728 – 9 has at least two real roots. -

16.J. Show that every polynomial of odd degree and real coefficients has a real root. Show that the polynomial p(x) = x¢ + 728 – 9 has at least two real roots. -

Advanced Engineering Mathematics
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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exercise 16.J

Please prove EVERYTHING, (if they affirm something, please prove it even if it seems easy)

16.J. Show that every polynomial of odd degree and real coefficients has a real root. Show that the polynomial p(x) = x* + 7x³ – 9 has at least two real roots. 16.K. If c > 0 and n is a natural number, there exists a unique positive num- ber b such that b* с. 16.L. Let f be continuous on I to R with f(0) < 0 and f(1) > 0. If N = {x € I: f(x) < 0}, and if c 16.M. Let f be a continuous function on R to R which is strictly increasing in the sense that if x' < x" then f(x') < f(x"). Prove that f is one-one and that its inverse function f- is continuous and strictly increasing. sup N, show that f(c) 0.
Transcribed Image Text:16.J. Show that every polynomial of odd degree and real coefficients has a real root. Show that the polynomial p(x) = x* + 7x³ – 9 has at least two real roots. 16.K. If c > 0 and n is a natural number, there exists a unique positive num- ber b such that b* с. 16.L. Let f be continuous on I to R with f(0) < 0 and f(1) > 0. If N = {x € I: f(x) < 0}, and if c 16.M. Let f be a continuous function on R to R which is strictly increasing in the sense that if x' < x" then f(x') < f(x"). Prove that f is one-one and that its inverse function f- is continuous and strictly increasing. sup N, show that f(c) 0.
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