Master The Second Derivative Test To Identify Local Minima And Maxima In Functions

By the 2nd derivative test, a point is a relative maximum at x = c if

f'(c) = 0 and f” < 0

the second derivative of the function evaluated at x = c is negative, and a relative minimum at x = c if the second derivative of the function evaluated at x = c is positive.

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. The first step is to find the critical points of the function, which are the points where the first derivative equals zero or where the function is not differentiable.

Once we have identified the critical points, we can use the second derivative test to determine if the point is a relative maximum or a relative minimum. The second derivative test involves taking the second derivative of the function and evaluating it at the critical point. If the second derivative is positive, then the point is a relative minimum, and if the second derivative is negative, then the point is a relative maximum.

However, it is important to note that if the second derivative evaluated at the critical point is zero, the test is inconclusive, and further analysis is required to determine whether the point is a maximum, minimum or neither.

In summary, the second derivative test is a useful tool for determining the nature of the critical points of a function and helping to identify local minimums and maximums.

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