The correct statement about the average rate of change over the interval \([a,b]\) for the function \(f(x)\) is:
The average rate of change is negative if \(f(b) < f(a)\).
Let's clarify the other statements:
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The average rate of change can be calculated using the formula \(\frac{b-a}{f(b)-f(a)}\): This statement is incorrect. The correct formula for the average rate of change of \(f\) from \(a\) to \(b\) is given by: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
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The average rate of change cannot be zero: This statement is incorrect. The average rate of change can be zero if \(f(b) = f(a)\), meaning there is no change in the function values over the interval.
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The average rate of change can be calculated using the formula \(\frac{f(b)-f(a)}{a-b}\): This statement is incorrect in its form. The average rate of change does use \(f(b) - f(a)\), but it divides by \(b - a\), not \(a - b\). A division by \(a - b\) would just flip the sign of the result.
Therefore, the only accurate statement here is that the average rate of change is negative if \(f(b) < f(a)\).
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