Answer
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The Mean Value Theorem (MVT) in calculus has three important corollaries:
1. Corollary 1: If a function \( f(x) \) is continuous on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists at least one point \( c \) in \((a, b)\) such that the instantaneous rate of change (derivative) of the function at \( c \) equals the average rate of change of the function over the interval \([a, b]\).
2. Corollary 2: If two functions \( f(x) \) and \( g(x) \) are continuous on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), and if \( g'(x) \neq 0 \) for all \( x \) in \((a, b)\), then there exists at least one point \( c \) in \((a, b)\) such that the derivative of the quotient \( \frac{f(x)}{g(x)} \) is equal to the quotient of the derivatives \( \frac{f'(c)}{g'(c)} \).
3. Corollary 3: If a function \( f(x) \) is continuous on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), and if \( f'(x) = 0 \) for all \( x \) in \((a, b)\), then \( f(x) \) is constant on the interval \([a, b]\).
These corollaries extend the insights provided by the Mean Value Theorem and have practical applications in analyzing functions and their behavior over intervals.
