Discovering Relative Max Via Second Derivative Test

2nd Derivative Test

f'(x) = 0 find f(x)TABLE for frelative extremas biggest = minsmallest = max 2nd Dervi only tells us concavity and points of inflection

The second derivative test is a mathematical tool used to determine whether a critical point of a function is a local maximum, local minimum, or saddle point.

The steps to apply the second derivative test are:
1. Find the critical points of the function by setting its first derivative to zero and solving for x.
2. Calculate the second derivative of the function.
3. Plug the critical points found in step 1 into the second derivative. This will give us the concavity of the function at the critical point.
4. If the result obtained in step 3 is positive, the function has a local minimum at that critical point.
5. If the result obtained in step 3 is negative, the function has a local maximum at that critical point.
6. If the result obtained in step 3 is zero, the test is inconclusive, and we must use other methods to determine the nature of the critical point.

It is important to note that the second derivative test only applies to functions that are twice differentiable at the critical point and that the test only tells us about local extrema, not global extrema. In addition, the second derivative test is not foolproof and should be used in conjunction with other methods to analyze a function.

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