Question

How does the function increase and decrease?

a) The function increases when the input variable is positive and decreases when it's negative.
b) The function increases when the input variable is negative and decreases when it's positive.
c) The function increases when the derivative is negative and decreases when the derivative is positive.
d) The function increases at a constant rate regardless of the input variable.

Answer 1

C. A function increases when its derivative is positive and decreases when its derivative is negative.

Step-by-step explanation:

When the input variable is positive, the function's output increases as we move along the number line. Conversely, when the input variable is negative, the function's output decreases as we move in the opposite direction along the number line. This relationship holds true for many functions, such as linear, quadratic, exponential, and trigonometric functions.

The function increases or decreases depending on the sign of its derivative, not directly on the input variable itself. When the value of the derivative of a function is positive, the function is increasing. Conversely, when the derivative is negative, the function is decreasing. The correct answer to how a function's behavior changes is related to the sign of its derivative is answer c: The function increases when the derivative is positive and decreases when the derivative is negative. None of the provided references to impulses, slopes, or velocities directly answer the student's question as they appear to be from different contexts.