Question

If a function is increasing, then its derivative is:

• A. Negative
• B. Zero
• C. Positive
• D. Undefined

Hint:

Think of the derivative as the "speedometer" of the function's height. If the speed is positive, you are moving up!

Answer 1

The Correct Option is C

To determine the derivative's behavior when a function is increasing, we need to understand the mathematical concept of differentiation and how it relates to the increase and decrease of functions.

Understanding Increasing Functions:

• A function \(f(x)\) is said to be increasing on an interval if, for any two points \(x_1\) and \(x_2\) such that \(x_1 < x_2\), we have \(f(x_1) < f(x_2)\).
• This means the function's output or value is going up as the input increases.

Role of Derivative:

• The derivative of a function, denoted by \(f'(x)\), represents the rate of change of the function with respect to \(x\).
• If \(f(x)\) is increasing on an interval, the rate at which \(f(x)\) changes is positive—that is, \(f'(x) > 0\) on that interval.

Explanation for the Correct Answer:

• Negative: If the derivative \(f'(x)\) were negative, it would indicate that the function is decreasing, not increasing. Thus, this cannot be the correct choice.
• Zero: A derivative of zero indicates that the function is neither increasing nor decreasing at that point—it could be a flat spot or a local extremum (maxima or minima). Hence this is not the correct choice for a generally increasing function.
• Positive: A positive derivative means the function's value is rising as the input increases, which matches the definition of a strictly increasing function. This is the correct choice.
• Undefined: A derivative is undefined where the function might have points of discontinuity, cusps, or vertical tangents, but this does not provide information about whether the function is generally increasing or decreasing over an interval.

Therefore, when a function is increasing, its derivative is positive.