Find the absolute extrema of the function on the closed interval. \[ y=2-|t-2|, \quad[-8,3] \] minimum \( (t, y)=(\square) \) maximum \( (t, y)=(\square) \)

To find the absolute extrema of the function \( y = 2 - |t - 2| \) on the interval \([-8, 3]\), we first need to determine the critical points and evaluate the function at the endpoints of the interval. 1. **Critical Points**: The function \( y = 2 - |t - 2| \) is piecewise defined based on the value of \( t \): - For \( t < 2 \): \( y = 2 - (2 - t) = t \) - For \( t \geq 2 \): \( y = 2 - (t - 2) = 4 - t \) We observe that at \( t = 2 \), the function changes from \( t \) to \( 4 - t \). 2. **Endpoints**: We evaluate \( y \) at the endpoints of the interval. - At \( t = -8 \): \[ y = 2 - |-8 - 2| = 2 - 10 = -8 \] - At \( t = 3 \): \[ y = 2 - |3 - 2| = 2 - 1 = 1 \] 3. **Critical Point**: Next, we evaluate the function at the critical point \( t = 2 \): \[ y = 2 - |2 - 2| = 2 - 0 = 2 \] 4. **Comparing Values**: - \( y(-8) = -8 \) - \( y(2) = 2 \) - \( y(3) = 1 \) Now we can identify the absolute maximum and minimum values: - The minimum value occurs at \( t = -8 \) with \( y = -8 \). - The maximum value occurs at \( t = 2 \) with \( y = 2 \). Thus, the extrema are: minimum \( (t, y) = (-8, -8) \) maximum \( (t, y) = (2, 2) \)