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let f(x)= -x^2 +3x on the interval [1,3] . Find the absolute maximum and absolute minimum of f(x) on this interval

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The absolute maximum and absolute minimum of implicit equations are difficult to find because of the uncertainty of the number of local maxima and minima.

For a quadratic equation, we know that there is only one single maximum/minimum which will automatically be the global value. If a global maximum falls within the interval, the smaller value of each of the limits will give the minimum within the interval.

Thus, interval = [1,3].
f(x)= -x^2 +3x
f'(x)=-2x+3
f'(x)=0 at x=1.5, falls within [1,3]
f"(x)=-2 f'(1.5) is a maximum.
Thus the maximum is at x=1.5, or f(1.5)=2.25.
The minimum is one of the two following values (evaluated at the limits of the given interval).
f(1) or f(3).

Can you take it from here?

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