Minima and Maxima of Quadratics and Maxima of Reciprocals of Quadratics

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To find the minimum or maximum of a quadratic we complete the square expressing the function in the form

Ifthe minimum will be wheresoand the minimum is at

Ifthe maximum will be wheresoand the maximum is at

For example, to find the minimum of

complete the square to getthen the minimum is at

To find the maximum ofcomplete the square to get then the maximum is at

We might also have to find the maxima of reciprocal quadratics such as

The quadratic here can have no roots if it is to have a maximum, or else at those roots we would havewhich has no value, and close to those roots the graph would tend to As before we complete the square to getTo maximise y we have to minimise the denominator ie minimiseThis has a minimum athencehas a maximum atThis is illustrated below. If the numerator were negative we would follow the same procedure, completing the square but now find a minimum, in this case at

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