Answer: Choice B is correct.
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Explanation:
We are told that the derivative function f ' (x) is positive for all real numbers x. This means that f(x) is increasing for all real numbers. An increasing function goes uphill as you move from left to right. This rules out choices C and D.
We're also told that the integral of f(t), from t = 4 to t = 7, is equal to 0. Because f(x) is increasing, this must mean that some portion of f(x) is below the x axis and some portion is above the x axis when focused on the interval of [tex]4 \le x \le 7[/tex]. This allows us to rule out choice E since there are no portions that are below the x axis here.
At this point, the answer is between A and B. We can rule out choice A for similar reasoning we did with choice E. This time, there is no portion that is above the x axis. We need a negative area (below the x axis) to counterbalance with a positive area (above the x axis) so that the two areas cancel out and add to 0. That doesn't happen with choice A.
The only thing left is choice B so that must be the answer. We have a portion below the x axis and a portion above it, so it's possible that these two areas cancel out to add to 0. Since we don't know the actual f(x) function, it's impossible to tell whether that happens or not.
