## Step 1: Understanding the Concept The derivative of a function at a point describes the instantaneous rate of change (or slope) of the function at that point. If the derivative is zero over an entire interval, this means that the rate of change is zero everywhere on that interval. ## Step 2: Analyzing the Consequences If the rate of change is zero across an interval, the function does not increase or decrease anywhere within that interval. Mathematically, if \( f'(x) = \) for all \( x \) in some interval, then the function \( f(x) \) must have the same value for every \( x \) in that interval. This is rooted in the Mean Value Theorem, which implies that a zero derivative over an interval indicates a horizontal (flat) function—i.e., it is constant. ## Step 3: Evaluating the Options Let's examine the options: - A. Increasing: Would require a positive derivative. - B. Decreasing: Would require a negative derivative. - C. Constant: Would require a derivative of zero, which matches our analysis. - D. Discontinuous: Discontinuity is unrelated here, as the question is about the derivative being zero. ## Final Answer **The correct answer is: C. Constant.** When the derivative of a function is zero over an interval, the function is constant on that interval.