91Ó°ÊÓ

Derivatives are essential tools in calculus, used to determine the rate at which a quantity changes. In simple terms, they tell us how a function behaves as its input changes.
When we model a function like a drug dosage response, the first derivative (\( f'(x) \)) gives us critical insights into how the response changes with dosage. By applying the derivative, we can determine spots where the function rises or falls, helping identify potential maximum and minimum points.

The process begins with the function \( f(x) = x^2(3-x) \). Using the product rule, which is versatile for functions that are products of simpler parts, we find that \( f'(x) = 6x - 3x^2 \). The first derivative reveals how the function's value changes with each incremental change in \( x \). This expression — \( 6x - 3x^2 \) — is a polynomial that helps locate critical points where the slope of the curve might change, indicating potential turning points or peaks and valleys in the curve.

Critical points are prominent features in the graph of a function where the derivative is zero or undefined. They are significant because they identify where a function’s rate of change pauses, usually signifying peaks and troughs.
From the derivative \( f'(x) = 6x - 3x^2 \), we set it to zero to find these points. Solving \( 6x - 3x^2 = 0 \) simplifies to \( x(2-x) = 0 \), giving the critical points \( x = 0 \) and \( x = 2 \).
These critical values hint where the function \( f(x) \) might have a local maximum or minimum.

Use sign diagrams to assess the intervals around these critical points:

• For \( 0 < x < 2 \), because \( f'(x) \) is positive, \( f(x) \) increases.
• For \( x > 2 \), \( f'(x) \) becomes negative, signaling that \( f(x) \) decreases.

This analysis helps identify \( x = 2 \) as a local maximum, making it a valuable point for understanding the function's behavior.

Inflection points are locations on a graph where the curve changes its concavity. Unlike critical points, which concern turning points in the rate of change, inflection points indicate where a function shifts from curving upwards to downwards or vice versa.

To determine inflection points, you need the second derivative. From the first derivative, \( f'(x) = 6x - 3x^2 \), we derive the second derivative, \( f''(x) = 6 - 6x \). By setting \( f''(x) = 0 \), we solve for \( x \): this results in \( x = 1 \).

• For \( x < 1 \), \( f''(x) > 0 \), indicating the curve is concave up.
• For \( x > 1 \), \( f''(x) < 0 \), showing a transition to concave down.

This change in sign at \( x = 1 \) confirms an inflection point, a critical feature for graphing as it depicts a natural shift in the curve's architecture.