Using the Second Derivative Test to Identify Local Minimums in Calculus

Second Derivative Test: f(x) has a local Minimum at x = c when f”(. c) is

positive

positive.

The Second Derivative Test is a useful tool in calculus that helps determine the nature of critical points on a function by analyzing its second derivative. Specifically, it can be used to determine whether a critical point corresponds to a local minimum or maximum.

To apply the Second Derivative Test, we need to follow these steps:

1. Find the critical points of the function f(x) by setting its first derivative f'(x) equal to zero and solving for x.

2. Calculate the second derivative f”(x) of the function.

3. Plug the critical points obtained in step 1 back into the second derivative function f”(x) and evaluate f”(c) for each critical point c.

4. Analyze the value of f”(c) for each critical point c to determine the nature of the critical point.

Now, let’s focus on the situation where f”(c) is positive.

If f”(c) is positive, it means that the concavity of the function changes from concave downward to concave upward at x = c. This implies that the function has a local minimum at x = c.

Why is this the case? When the second derivative, f”(x), is positive at a critical point x = c, it indicates that the slope of the tangent line to the graph of f(x) is increasing as we move from left to right around x = c. In other words, the graph is “bending upward” around the critical point.

This behavior leads to the function having a local minimum at x = c. Intuitively, it means that the function reaches its lowest point at c and starts increasing again on both sides. Mathematically, this can be confirmed by observing that the derivative, f'(x), changes from negative to positive as we move from left to right through x = c.

Hence, when f”(c) is positive, it indicates that f(x) has a local minimum at x = c. This information can be useful when analyzing the behavior and characteristics of a function.

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