Determining Local Extrema with the Second Derivative Test in Math

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

negative

negative.

The Second Derivative Test is a method used to determine whether a critical point of a function is a local maximum or a local minimum. To apply the test, we need to consider the second derivative of the function at the critical point.

Let’s assume that f(x) is a twice differentiable function and c is a critical point where f'(c) = 0 or f'(c) does not exist. To determine if f(x) has a local maximum at x = c, we need to check the sign of the second derivative, f”(c).

If f”(c) is negative, then f(x) has a concave down shape at x = c. This means that the graph of the function is decreasing around the critical point. In other words, the function is curving downwards at x = c.

Since the function is decreasing, it reaches a maximum value at x = c. This maximum value is known as a local maximum since it is higher than the values of f(x) on either side of x = c within a certain interval.

On the other hand, if f”(c) is positive, then f(x) has a concave up shape at x = c. This means that the graph of the function is increasing around the critical point. In this case, the function is curving upwards at x = c.

Since the function is increasing, it reaches a minimum value at x = c. This minimum value is known as a local minimum since it is lower than the values of f(x) on either side of x = c within a certain interval.

To summarize, if the second derivative f”(c) is negative, then f(x) has a local maximum at x = c. If the second derivative f”(c) is positive, then f(x) has a local minimum at x = c.

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