# 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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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