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The second derivative, denoted as $ f’’(x) $ or $ \frac{d^2y}{dx^2} $ , provides crucial information about the shape and behavior of a function. It helps us understand concavity and inflection points.

Concavity

The second derivative reveals the concavity of a function.

• ** $ f’’(x) > 0 $ on an interval:** The function is concave up (or “holds water”) on that interval. The graph curves upwards like a smile.

• ** $ f’’(x) < 0 $ on an interval:** The function is concave down (or “spills water”) on that interval. The graph curves downwards like a frown.

• ** $ f’’(x) = 0 $ on an interval:** This doesn’t automatically mean the function is neither concave up nor concave down. Further investigation is needed. Test for Concavity

y = x^3 - 3x
y = x^2
y = -x^2

Inflection Points

An inflection point is a point on the graph where the concavity changes. This means the function transitions from concave up to concave down, or vice versa. To find inflection points:

1. Find the second derivative: $ f’’(x) $ .
2. Solve for $ f’’(x) = 0 $ or where $ f’’(x) $ is undefined. These are potential inflection points.
3. Test the concavity on intervals around the potential inflection points. If the concavity changes across a potential inflection point, it is an inflection point. If it doesn’t change, it’s not an inflection point. Second Derivative Test for Inflection Points Example:

Let’s consider the function $ f(x) = x^3 - 3x $ .

1. $ f’(x) = 3x^2 - 3 $
2. $ f’’(x) = 6x $
3. $ f’’(x) = 0 $ when $ x = 0 $ .
4. Testing intervals:
   • $ x < 0 $ : $ f’’(x) < 0 $ (concave down)
   • $ x > 0 $ : $ f’’(x) > 0 $ (concave up)

Since the concavity changes at $ x = 0 $ , there is an inflection point at $ (0, f(0)) = (0, 0) $ .

Relationship to the First Derivative and Optimization Extrema and Concavity

The second derivative can be used in conjunction with the first derivative to determine the nature of critical points (where $ f’(x) = 0 $ or $ f’(x) $ is undefined):

• Second Derivative Test: If $ f’(c) = 0 $ and:
  • $ f’’(c) > 0 $ , then $ f(c) $ is a local minimum.
  • $ f’’(c) < 0 $ , then $ f(c) $ is a local maximum.
  • $ f’’(c) = 0 $ , the test is inconclusive; further investigation is needed (e.g., the First Derivative Test).

Applications

Understanding concavity and inflection points is crucial for:

• Sketching graphs: Accurately depicting the shape of a function.
• Optimization problems: Identifying maximum and minimum values.
• Modeling real-world phenomena: Analyzing rates of change and trends.

This rundown provides a foundation for analyzing functions using the second derivative in Calculus AB. Remember to always consider the context of the problem and use appropriate tests to draw accurate conclusions.