Certainly! Let's analyze the function [tex]\( f(t) = |t - 10| \)[/tex] over the interval [tex]\( 7 \leq t \leq 16 \)[/tex].
### Step-by-Step Solution
1. Definition of the Function:
The function [tex]\( f(t) = |t - 10| \)[/tex] represents the absolute value of [tex]\( t \)[/tex] minus 10. Absolute value functions typically form a "V" shape.
2. Critical Points:
The absolute value function [tex]\( |t - 10| \)[/tex] reaches its minimum when [tex]\( t = 10 \)[/tex]. At this point, the value of the function is:
[tex]\[
f(10) = |10 - 10| = 0
\][/tex]
So, [tex]\( t = 10 \)[/tex] is a critical point within the interval.
3. Endpoints of the Interval:
We need to evaluate the function at the endpoints of the interval [tex]\( t = 7 \)[/tex] and [tex]\( t = 16 \)[/tex].
[tex]\[
f(7) = |7 - 10| = 3
\][/tex]
[tex]\[
f(16) = |16 - 10| = 6
\][/tex]
4. Finding the Absolute Max and Min:
- The minimum value of [tex]\( f(t) \)[/tex] over the interval occurs at [tex]\( t = 10 \)[/tex] with [tex]\( f(10) = 0 \)[/tex].
- The maximum value of [tex]\( f(t) \)[/tex] will be at the endpoint where [tex]\( f(t) \)[/tex] is largest: comparing [tex]\( f(7) = 3 \)[/tex] and [tex]\( f(16) = 6 \)[/tex], the maximum value occurs at [tex]\( t = 16 \)[/tex] with [tex]\( f(16) = 6 \)[/tex].
### Summary of Results
- The absolute maximum value is [tex]\( 6 \)[/tex].
- The absolute maximum value occurs at [tex]\( t = 16 \)[/tex].
- The absolute minimum value is [tex]\( 0 \)[/tex].
- The absolute minimum value occurs at [tex]\( t = 10 \)[/tex].
### Graph and Labels
To visualize this, you can draw the graph of [tex]\( f(t) = |t - 10| \)[/tex] from [tex]\( t = 7 \)[/tex] to [tex]\( t = 16 \)[/tex]. This function will have a "V" shape with its vertex at [tex]\( (10, 0) \)[/tex]. The graphs will rise to the endpoints:
[tex]\[
(7, 3) \quad \text{and} \quad (16, 6)
\][/tex]
Here’s a plot label of points where the absolute extrema occur, and include their coordinates:
- Minimum at (10, 0)
- Maximum at (16, 6)
### Final Answer Entries
The absolute maximum value is [tex]\( \boxed{6} \)[/tex].
The absolute maximum value occurs at [tex]\( \boxed{16} \)[/tex].
The absolute minimum value is [tex]\( \boxed{0} \)[/tex].
The absolute minimum value occurs at [tex]\( \boxed{10} \)[/tex].
This completes the analysis and provides a clear understanding of the absolute extrema for the given function within the interval.
