Please provide a formal mathematical proof for both the statements and answer the following question. Thank you. 37. Suppose that a and b are two consecutive roots of a polynomial function f, with a < b.Suppose a and b are non-repeated roots. Consequently, f(x) = (x – a)(x – b)g(x) forsome polynomial function g. Consider the statements:(I) g(a) and g(b) have opposite signs(II) f'(x) = 0 for some r € (a, b)Of these statements,A. Both I and II are trueB. Only I is trueC. Only II is trueD. Both I and II are false

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37. Suppose that a and b are two consecutive roots of a polynomial function f, with a < b. Suppose a and b are non-repeated roots. Consequently, f(r) = (x – a)(x – b)g(x) for some polynomial function g. Consider the statements: (I) g(a) and g(b) have opposite signs (II) f'(x) = 0 for some r € (a, b) Of these statements, A. Both I and II are true B. Only I is true C. Only II is true D. Both I and II are false

37. Suppose that a and b are two consecutive roots of a polynomial function f, with a < b. Suppose a and b are non-repeated roots. Consequently, f(r) = (x – a)(x – b)g(x) for some polynomial function g. Consider the statements: (I) g(a) and g(b) have opposite signs (II) f'(x) = 0 for some r € (a, b) Of these statements, A. Both I and II are true B. Only I is true C. Only II is true D. Both I and II are false

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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Please provide a formal mathematical proof for both the statements and answer the following question. Thank you.

37. Suppose that a and b are two consecutive roots of a polynomial function f, with a < b.
Suppose a and b are non-repeated roots. Consequently, f(x) = (x – a)(x – b)g(x) for
some polynomial function g. Consider the statements:
(I) g(a) and g(b) have opposite signs
(II) f'(x) = 0 for some r € (a, b)
Of these statements,
A. Both I and II are true
B. Only I is true
C. Only II is true
D. Both I and II are false

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