The Second Derivative Test

The Second Derivative Test is a method used in calculus to classify critical points of a function as local maximums, local minimums, or neither. It utilizes the value of the function's second derivative at a critical point.

Its main purpose is to help identify and distinguish between local extrema.

The test works by examining the concavity of the function at a critical point:

• If the second derivative f''(x) > 0 at a critical point, the function is concave up at that point, indicating a local minimum.
• If the second derivative f''(x) < 0 at a critical point, the function is concave down at that point, indicating a local maximum.

It relates the rate of change of the slope (concavity) to whether the function is at a "bottom" (concave up) or a "top" (concave down) in that vicinity.

Follow these steps to use the Second Derivative Test:

1. Find the first derivative, f'(x).
2. Find the critical points by setting f'(x) = 0 or finding points where f'(x) is undefined (but the original function is defined). *Note: The Second Derivative Test typically works best for critical points where f'(c) = 0.*
3. Find the second derivative, f''(x).
4. Evaluate the second derivative at each critical point 'c' found in step 2.
5. Apply the Second Derivative Test conclusion based on the sign of f''(c):
   • If f''(c) > 0, then f(c) is a **local minimum**.
   • If f''(c) < 0, then f(c) is a **local maximum**.
   • If f''(c) = 0, the test is **inconclusive**.
6. If the test is inconclusive, you must use the First Derivative Test to classify that critical point.
7. Evaluate the original function f(x) at the critical points identified as local extrema to find the actual local maximum or minimum *values*.

The Second Derivative Test is inconclusive when you evaluate the second derivative at a critical point 'c' and find that f''(c) = 0. In this situation, the test does not provide enough information to determine if the point is a local maximum, a local minimum, or neither (it could be an inflection point, for example).

The test also doesn't apply directly to critical points where the first derivative f'(x) is undefined. For these points, you must rely on the First Derivative Test.

Both tests classify critical points, but they use different information:

• **First Derivative Test:** Looks at the *sign change* of f'(x) *around* a critical point. It tells you whether the function is switching from increasing to decreasing (max) or decreasing to increasing (min). It works for all critical points (where f'(x) = 0 or is undefined).
• **Second Derivative Test:** Looks at the *sign* of f''(x) *at* a critical point (specifically where f'(c) = 0). It tells you about the concavity, which implies whether it's a local max or min. It's often quicker computationally if f''(c) ≠ 0 but is inconclusive if f''(c) = 0 or if f'(c) was undefined.

You can use either test. Many prefer the First Derivative Test as it is universally applicable to all critical points. However, if the second derivative is easy to calculate and non-zero at the critical points, the Second Derivative Test can be faster.