Second Derivative Test. Worked Example Conclusion

University
Massachusetts Institute of Technology
Course
Multivariable Calculus
Pages

1

Academic year

2022
Author

RacingChronos
Views

58

p {margin: 0; padding: 0;} .ft00{font-size:20px;font-family:Arial;color:#000000;} .ft01{font-size:18px;font-family:ArialMT;color:#000000;} .ft02{font-size:18px;line-height:23px;font-family:ArialMT;color:#000000;} Second derivative test. Worked example conclusion Computing the second partial derivatives can help us determine what type of point itis. What is the derivative of this function with respect to x? If we set A equal to 2 over x cubed y, B will be 1 and C will be 2. This tells us that ACminus B squared is equal to 4 minus 1. AC minus B squared is 2 times 2 minus 1squared, which equals 3. This tells us that we are either at a local minimum or a localmaximum. We can see that A is positive and thus, it is a local minimum. In fact, it is the globalminimum. The maximum is not at a critical point. It's on the boundary or at infinity, so we needto check what happens when x and y go to 0 as both approaches infinity. If x or ygoes to infinity, then the function goes to infinity; also if x or y goes to 0, then oneover xy goes to infinity. The function goes to infinity when x goes to infinity or y goesto infinity or x and y go to 0. So it's not at a critical point. In general, we have to check for both critical points and boundaries to decide whathappens.