AHL 5.10—Second derivatives, testing for max and min Notes

The Second Derivative

Definition and Notation

The second derivative is a measure of how the rate of change of a function is itself changing, obtained by differentiating a function twice. There are two common notations for the second derivative:

1. Leibniz notation: $\frac{d^2y}{dx^2}$
2. Lagrange notation: $f''(x)$

For example, if we have a function $f(x) = x^3$, its first derivative is $f'(x) = 3x^2$, and its second derivative is $f''(x) = 6x$.

Graphical Interpretation

The second derivative provides crucial information about the shape and behavior of a function's graph:

1. Concavity: The sign of the second derivative determines the concavity of the function.
   1. if $f''(x) > 0$, the function is concave up (shaped like a cup).
   2. If $f''(x)< 0$, the function is concave down (shaped like an inverted cup).
2. Inflection Points: Points where the concavity changes (i.e., where $f''(x) = 0$ or is undefined) are called inflection points. Because $\frac{dy}{dx}$ is at a maximum or minimum at this point, the tangent line will change from increasing to decreasing—concave down—or change from decreasing to increasing—concave up —hence why they are studied.

Relationship Between f, f', and f''

Understanding the relationships between a function and its derivatives is crucial for graphical analysis:

1. f and f':
   1. When $f'(x) < 0$, f(x) is decreasing
   2. When $f'(x) = 0$, f(x) has a horizontal tangent.
2. f' and f'':
   1. When $f''(x)> 0$, f'(x) is increasing (f(x) is concave up).
   2. When $f''(x)< 0$, f'(x) is decreasing (f(x) is concave down).
   3. When $f''(x) = 0$, f'(x) has a horizontal tangent (potential inflection point for f(x)).

Using Technology for Graphical Analysis

Modern graphing calculators and software explore these relationships numerically.

1. Graphing: Plot f(x), f'(x), and f''(x) on the same coordinate system to visualize their relationships.
2. Numerical Differentiation: Use technology to calculate derivatives when analytical methods are challenging.
3. Dynamic Exploration: Use sliders or animation features to see how changes in parameters affect the graphs of f, f', and f''.

Testing for Local Maximum and Minimum Points

• Local maximum and minimum points are where a function reaches its highest or lowest value within a small interval.
• These points are also called stationary points because the derivative of the function is zero at these locations.

Steps to Find and Test Local Maximum and Minimum Points

1. Find the Derivative: Calculate the first derivative, $f'(x)$.
2. Identify Stationary Points: Solve $f'(x) = 0$ to find potential maximum or minimum points.
3. Use the Second Derivative Test:
4. Calculate the second derivative, $f''(x)$.
5. Evaluate $f''(x)$ at each stationary point:
   • If $f''(x) > 0$, the point is a local minimum (the curve is concave up).
   • If $f''(x) < 0$, the point is a local maximum (the curve is concave down).
   • If $f''(x) = 0$, the test is inconclusive. Consider using other methods, such as the first derivative test.