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?

neela | Student

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



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=>


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