College Math Placement Practice Test

If a function is increasing, what can be inferred about its derivative?

1. The derivative is negative

2. The derivative is zero

3. The derivative is positive

4. The derivative is undefined

The correct answer is: The derivative is positive

When a function is described as increasing, it means that as you move from left to right along the x-axis, the values of the function rise. In terms of rate of change, this behavior is directly linked to the function's derivative. The derivative of a function represents its rate of change at any given point. If the function is increasing, this means that wherever you look on the graph of the function, the slope of the tangent line (which is what the derivative measures) is positive. Therefore, the derivative must be positive for all points in the interval where the function is increasing. This positive derivative indicates that for any small increase in x, the function's value also increases, which is the defining trait of an increasing function. Hence, the correct inference about the derivative when a function is increasing is that it is positive.