We learnt about the location of roots of quadratic equation in class today. I have a problem in one case specifically, the one where both the roots are required in the interval $(k_1,k_2)$ where $k_1,k_2$ are real constant numbers. So the conditions given to us were,
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$\Delta$ greater than or equal to $0$ (fairly obvious)
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$af(k_1)>0$ where $f(k_1)$is the value of the function at $k_1$ and $a$ is the leading coefficient
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$af(k_2)>0$ (both of these are again understandable)
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then comes $k_1<-b/2a<k_2$ i.e. the abscissa of the vertex of the parabola lies in the interval required.
My question or rather statement is that the 4th condition is not necessary, as 2nd and 3rd conditions already make it clear that the vertex is going to lie between the endpoints of the interval.
But the interesting part is that this condition is necessary. Can't seem to find the proof rather than straight-up statements. Help me….. and sorry for the formatting..
Best Answer
You're missing an essential point: conditions $2$ and $3$ mean that $k_1$ and $k_2$ are outside the interval of the roots. However, they might be on the same side of this interval. Therefore, supposing conditions $2$ and $3$ is not enough, we must add the condition that $k_1, k_2$ are not on the same side, which is equivalent to their arithmetic mean being in the interval $(k_1,k_2)$.