Pregunta
Use a graph to determine the local minimum(s) of
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The local minima of
Solución
To determine the local minima of the piecewise function
let’s analyze each interval and the behavior of the function at the transition points.
1. Interval
- Graph Shape: V-shaped with the vertex at
. - Local Minimum: At
, . This is the lowest point in this interval.
2. Interval
- Graph Shape: Increasing function.
- Behavior at
: (continuous with the previous interval). - Derivative from Left (
): . - Derivative from Right (
): .
- Derivative from Left (
- Conclusion:
is not a local minimum since the function continues to increase on both sides.
3. Interval
- Graph Shape: Decreasing linear function.
- Behavior at
: . - Derivative from Left (
): . - Derivative from Right (
): .
- Derivative from Left (
- Conclusion:
is a local maximum since the function changes from increasing to decreasing.
4. Endpoint
- Value:
. - Behavior: As
approaches 8 from the left, the function decreases towards . - Conclusion:
is a local minimum because it is the lowest point in its immediate neighborhood.
Summary of Local Minima:
-
: Local minimum from the first interval. -
: Local minimum at the endpoint.
Therefore, the only local minima of
Answer:
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The local minimum of the piecewise function
To visually confirm the local minima, sketch the three components of the piecewise function together. The first segment forms the V-shape with its bottom at (0,0). In the second segment,
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