Question

find the absolute maximum and minimum values of the following function on the given interval. then graph the function.
f(t)=2 - |t|, -1 ≤ t ≤ 7
find the absolute maximum value. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the absolute maximum value 2 occurs at t = 0.
(simplify your answers. use a comma to separate answers as needed.)
b. there is no absolute maximum.
find the absolute minimum value. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the absolute minimum value □ occurs at t = □.
(simplify your answers. use a comma to separate answers as needed.)
b. there is no absolute minimum.

Explanation:

Step1: Analyze the function \(f(t)=2 - |t|\)

The function \(y = |t|\) is a V - shaped graph. The function \(f(t)=2-|t|\) is a transformation of \(y = |t|\), it is reflected over the \(t\) - axis and then shifted up by 2 units. The vertex of \(y = |t|\) is at \((0,0)\), so the vertex of \(f(t)=2 - |t|\) is at \((0,2)\).

Step2: Evaluate the function at the endpoints of the interval

We are given the interval \(-1\leq t\leq7\).

• When \(t=-1\), \(f(-1)=2-|-1|=2 - 1=1\)
• When \(t = 7\), \(f(7)=2-|7|=2 - 7=-5\)

Answer:

The absolute minimum value \(-5\) occurs at \(t = 7\).