Question

analyzing graphs which statement is true about the local minimum of the graphed function? over the interval 4, 7, the local minimum is −7. over the interval −1, 4, the local minimum is 0. over the interval −2, −1, the local minimum is 25. over the interval −4, −2, the local minimum is 0.

Explanation:

Step1: Analyze each interval

• For interval \([4, 7]\): The graph has a point \((5.1, -7)\) here, which is a local minimum (lowest in this interval).
• For interval \([-1, 4]\): The graph decreases then increases, but the minimum here isn't 0 (the roots are at \(x\)-intercepts, but the minimum in this interval is lower than 0).
• For interval \([-2, -1]\): The point \((0, 25)\) is a local maximum, not minimum here.
• For interval \([-4, -2]\): The graph is increasing, so the minimum is at \(x = -4\) (the left end), not 0.

Step2: Confirm the correct statement

Only the statement about \([4, 7]\) with local minimum \(-7\) matches the graph's behavior.

Answer:

Over the interval \([4, 7]\), the local minimum is \(-7\).