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Question

Show that the values of the function x2+34x71x2+2x7 do not lie between 5 and 9 for all, xR.

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Solution

Let x2+34x71x2+2x7=y,xR

x2+34x71=y(x2+2x7)

x2+34x71=yx2+2yx7y

(y1)x2+(2y34)x+717y=0

As xR discriminant must be greater than zero

(2y34)24(y1)(717y)

(y17)2(y1)(717y)

y2+28934y71y717y2+7y

8y2112y+3600

y214y+450

y25y9y+450

y(y5)9(y5)0

(y5)(y9)0

So y(,5][9,)

So it can be in (5,9)


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