Mastering The Second Derivative Test: Analyzing Function Behavior And Critical Points

Second Derivative Test (concavity, local extrema)

f'(c)=0 and f”(c)<0 (conc down) --> local max at x=cf'(c)=0 and f”(c)>0 (conc up) –> local min at x=c*if f”(c)=0 SDT cannot be used

The second derivative test is a mathematical tool used to analyze the behavior of a function based on the shape of its second derivative. It allows us to determine the concavity and identify the local extrema of the function.

To apply the second derivative test, we need to follow these steps:

1. Find the second derivative of the function:
Derivative of f(x) = f'(x)
Derivative of f'(x) = f”(x)

2. Set f”(x) = 0 and solve for x to find critical points of the function.

3. For each critical point found in Step 2, determine the concavity of the function:
– If f”(x) > 0, the function is concave up (convex) at this point.
– If f”(x) < 0, the function is concave down (concave) at this point. - If f''(x) = 0, we cannot use the second derivative test to determine the concavity. 4. Next, determine whether the critical point is a local maximum, local minimum, or neither by looking at the sign of f'(x) near the critical point: - If f'(x) changes sign from negative to positive as we move from left to right, then the critical point is a local minimum. - If f'(x) changes sign from positive to negative as we move from left to right, then the critical point is a local maximum. - If f'(x) does not change sign, then the critical point is neither a local maximum nor a local minimum. It's important to note that the second derivative test only applies to differentiable functions, meaning that the function must have a continuous second derivative. Overall, the second derivative test is a powerful tool to analyze the behavior and identify the critical points of a function.

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