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By R. J. Goult

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Extra info for Applicable Mathematics: A Course for Scientists and Engineers

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Show that, if A, B, C, are real and A> 0, then Ax 2 + 2Bx + C ;;;. 0 for all real values of x if, and only if, B2 - AC ~ 0. Apply this result to the expression 35 NUMBERS AND NUMBER SYSTEMS and derive the Cauchy-Schwarz inequality { ~ (a;b;)} ~{~at}{~ b{} 2 7. If ah a 2, ... , an and bt. b 2, ... ] 8. Establish whether the following numbers are rational or irrational: (a) 1 +y2 +y3, (b)y(7 +y7), (d)~3 +~2. 9. If z h z 2, z 3 , are any complex numbers such that z~ + z~ + z~ = z 1z 2 + z 2 z 3 + z 3 z 1 show that 10.

We may, however, make the result plausible by the following geometric argument. 2, OAC is a segment of a circle centre 0 and radius a, and OAB and OAD are right-angled triangles. a X 0 Clearly That is D Fig. 2 Area OAB

Show that, if A, B, C, are real and A> 0, then Ax 2 + 2Bx + C ;;;. 0 for all real values of x if, and only if, B2 - AC ~ 0. Apply this result to the expression 35 NUMBERS AND NUMBER SYSTEMS and derive the Cauchy-Schwarz inequality { ~ (a;b;)} ~{~at}{~ b{} 2 7. If ah a 2, ... , an and bt. b 2, ... ] 8. Establish whether the following numbers are rational or irrational: (a) 1 +y2 +y3, (b)y(7 +y7), (d)~3 +~2. 9. If z h z 2, z 3 , are any complex numbers such that z~ + z~ + z~ = z 1z 2 + z 2 z 3 + z 3 z 1 show that 10.

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