The second derivative test is a mathematical method used to classify critical points of a function, determining if they correspond to local minima, maxima, or saddle points. It relies on the sign of the second derivative: positive indicates a local minimum, negative indicates a local maximum, and zero means the test is inconclusive. The document also provides a practical example using the function f(x) = x³ - 6x² + 12x + 5 to illustrate the process of applying the test.

1. Second
Derivative Test

2. 01
What is the second
derivative test?

3. The Second Derivative Test is a powerful tool for understanding the behavior of a
function around critical points. It leverages the second derivative of a function, denoted
as f''(x). Here's the key connection:
What is the second derivative test?
● If f''(x) is positive at a critical point, it suggests that the function has a local
minimum at that point
● If f''(x) is negative, it implies a local maximum
● And if f''(x) equals zero, the test is inconclusive, indicating a possible saddle point
In essence, the Second Derivative Test tells you whether the function is bending upward
(concave up) or downward (concave down) at a critical point

4. Let's put the Second Derivative Test to work
with a real function. Consider this function:
f(x) = x3 - 6x2 + 12x + 5
Our mission is to find the critical points and
determine whether they represent local
minima, maxima, or saddle points
Applying the second derivative test

5. Step 1: find critical points
Set f'(x) equal to zero and solve for x to
find critical points
3x^2 - 12x + 12 = 0
Once you've solved for x, you'll have the
critical points
Begin by calculating the first and
second derivatives of f(x). The first
derivative, f'(x), gives us critical points:
f'(x) = 3x^2 - 12x + 12

6. Step 2: use the second derivative test
Calculate f''(c)
If f''(c) is positive, it
indicates a local minimum
at x=c
If f''(c) is negative, it
suggests a local
maximum
If f''(c) equals zero, the
test is inconclusive,
indicating a possible
saddle point
By applying this test to each critical point, you can determine the nature of the extrema

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