Question

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

Explanation:

Step1: Analyze each option

• Option 1: Over \([-4, -2]\), the function is increasing from \((-5,0)\) to \((0,25)\), so local maximum here is not 0.
• Option 2: Over \([-2, -1]\), the function is at the peak near \(x = 0\), local maximum is 25? Wait, no, \([-2,-1]\) is left of peak. Wait, re - check. The peak is at \((0,25)\). Over \([-2,-1]\), the function is still increasing towards the peak. But the option says local maximum is 25? Wait, no, let's check the graph. Wait, the key is to check the local minimum (wait, the question is about local minimum? Wait, the question says "local minimum"? Wait, no, the options are about local maximum? Wait, the original question: "Which statement is true about the local minimum of the graphed function?" Wait, maybe a typo, but let's check the options. Wait, the last option: Over \([4,7]\), the function has a minimum at \((5, - 7)\), so the local minimum there is - 7. Let's re - evaluate:
• Option 1: Over \([-4,-2]\), the function goes from \((-5,0)\) up, so no local minimum (it's increasing), and the option says local minimum is 0, wrong.
• Option 2: Over \([-2,-1]\), the function is increasing towards the peak, no local minimum, and the value 25 is the peak (local maximum), wrong.
• Option 3: Over \([-1,4]\), the function has a local maximum at \((0,25)\) and then decreases, but the option says local minimum is 0, wrong.
• Option 4: Over \([4,7]\), the function has a minimum at \(x = 5\) with \(y=-7\), so the local minimum over \([4,7]\) is - 7.

Answer:

Over the interval \([4, 7]\), the local minimum is -7 (the last option, e.g., if options are labeled as D: Over the interval \([4, 7]\), the local minimum is -7, then the answer is D. Over the interval \([4, 7]\), the local minimum is -7)