# `y = 3 - |t - 3|, [-1,5]` Find the absolute extrema of the function on the closed interval.

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### 1 Answer

Hello!

Consider this function for t in [-1, 3] and [3, 5] separately.

At the first interval t<=3 and |t-3| = -(t-3) = 3-t.

So y(t) = 3 - (3-t) = t.

The maximum is at the maximum t, i.e. at t=3, y(3) = 3.

The minimum is at the minimum t, i.e. at t=-1, y(-1) = -1.

At the second interval t>=3 and |t-3| = t-3.

So y(t) = 3 - (t-3) = 6 - t.

The maximum is at the minimum t, i.e. again at t=3 and with the same value 3.

The minimum is at the maximum t, i.e. at t=5, y(5) = 1.

As the whole, the greatest value of y is 3 (at t=3) and the smallest is -1 (at x=-1).

The answer: the global minimum is **-1 at t=-1** and the global maximum is **3 at t=3**.