The 2nd Derivative Test

The 2nd Derivative Test...so elegant it makes the 1st derivative test look like the ugly step-sister.  If you got one thing from the notes today, hopefully it was the connection between concavity and the 2nd derivative.

Recall:

• 2nd Derivative > 0, concave up
• 2nd Derivative < 0, concave down
• 2nd Derivative = 0 or DNE, point of inflection

In order to use the 2nd Derivative Test to prove points are indeed extrema you must do the following in you HW:

1. graph (use your graphing calculator)
2. find derivative, set equal to zero to locate critical points
3. find 2nd derivative & plug critical points into 2nd derivative
• If positive...then it's a minimum
• If negative...then it's a maximum
• If it's zero...use 1st derivative test (or see if it's a POI)
4. find increasing/decreasing intervals (use critical points to help identify intervals)
5. find concave up/concave down intervals (use POIs to help identify intervals)

Example 1: Find any local extrema of f(x) = x⁴ − 8 x² using the Second Derivative Test.

f′(x) = 0 at x = −2, 0, and 2. Because f″(x) = 12 x² −16, you find that f″(−2) = 32 > 0, and f has a local minimum at (−2,−16); f″(2) = 32 > 0, and f has local maximum at (0,0); and f″(2) = 32 > 0, and f has a local minimum (2,−16).

Example 2: Find any local extrema of f(x) = sin x + cos x on [0,2π] using the Second Derivative Test.

f′(x) = 0 at x = π/4 and 5π/4. Because f″(x) = −sin x −cos x, you find that and f has a local maximum at . Also, . and f has a local minimum at .