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fm(x)=x^2 +7(m-1)x+m-1 is a family of functions.Which are m values so that the curves...

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pavelpimen | Student, Undergraduate | Honors

Posted June 2, 2009 at 9:07 PM via web

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fm(x)=x^2 +7(m-1)x+m-1 is a family of functions.Which are m values so that the curves associated to the fm(x), have the vertex set above ox axis?

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neela | High School Teacher | Valedictorian

Posted June 3, 2009 at 2:53 AM (Answer #1)

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f(m,x) = x^2+7m-1)x+m-1 . We shall write the right side as below.

f(m,x)={x+7(m-1)/2}^2-[49(m-1)^2/2]^2-(m-1)]

={x+7(m-1)/2}^2-(1/2){(m-1)(49m-53)}.

The curve has its vertex at:

x=-7(m-1)/2 and its y=f(m,-7(m-1)/2)) = (-1/2){m-1)(49m-53} which is positive if(m-1)(49m-53) is negative=>

1<m<53/49.

Therefore, the vertices of the family of curves will be above x axis if  f(m,-7(m-1)/2) is positive  and this can happen only  when the values of m lie between 1 and 53/49

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