How To Perform The Second Derivative Test To Find Critical Points Of A Function: Steps And Guidelines

Second Derivative Test

D(x,y) = fxx(x,y) * fyy(x,y) – (fxy(x,y))^2 – D(x,y) > 0, fxx(x,y) > 0 => local maximum – D(x,y) > 0, fxx(x,y) < 0 => local minimum – D(x,y) < 0 => saddle point

The second derivative test is a method used to find the nature of a critical point of a function. It involves analyzing the sign of the second derivative of the function at the critical point. Here are the steps to perform the second derivative test:

1. Find the first derivative of the function.
2. Find the critical points of the function by solving for where the first derivative equals zero.
3. Find the second derivative of the function.
4. For each critical point, evaluate the sign of the second derivative at that point.
5. If the second derivative is positive at a critical point, then that point is a local minimum.
6. If the second derivative is negative at a critical point, then that point is a local maximum.
7. If the second derivative is zero at a critical point, then the second derivative test is inconclusive, and another method must be used to determine the nature of the critical point.
8. If there are no critical points, the function is either always increasing or always decreasing.

It is important to note that the second derivative test only works for finding the nature of local extrema. For finding the global extrema, the function must be analyzed at the endpoints of its domain or at any other points where the function is undefined.

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