Lecture Note
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
2
Academic year
2022
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Second derivative test: General case The second derivative test indicates that there is a critical point (x0, y0) of a functionof two variables of f, and then let's compute the partial derivatives. Let's call capital A the second derivative with respect to x. Let's call capital B thesecond derivative with respect to x and y, and let C equal f sub yy at this point. Oncewe have computed the second derivative, we plug in values of x and y at the criticalpoint. These numbers will just be numbers; now what we do is look at the quantityAC minus B squared. Don`t forget a 4; you will see why there isn't one. So if ACminus B squared is positive, then there's two sub-cases. If A is positive, then it's alocal minimum. In the second case, if AC minus B squared is positive but A is negative, then it'sgoing to be a local maximum. If AC minus B squared is negative, then it's a saddlepoint. And if AC minus B squared is 0, we don't know whether it's going to be aminimum, a maximum or a saddle. It's important to understand why order of operations can affect the outcome of amathematical equation. As a result, we will try to relate our old recipe with a newone.
Next, let us check that these two functions satisfy the same equation in the specialcase where the function is ax squared plus bxy plus cy squared. To find the secondderivative, we must take the derivative of the first partial with respect to x and addthat to the first partial with respect to x. First, we will take the derivative of 2ax plusby.So w sub xx will be - let's take the partial with respect to x again. That's 2a. w sub xy,I take the partial with respect to y, I will get b. OK, now we need also the partial withrespect to y. So w sub y is bx plus 2cy.In case you do not believe what we just saw about the mixed parcels with sub yx,well, you can check, and it's again b. So they are, indeed, the same thing. And w subyy will be 2c. So if we now look at these quantities, that tells us big A is 2 little a. BigB is little b. Big C is 2 little c, so AC minus B squared is what we used to call 4 littleac minus b squared. Now that you've seen the cases for local maxima, compare them. The first case iswhen either AC minus B squared is negative or 4AC minus B squared is negative.The second case is when capital AC minus B squared is positive, local and localmax corresponds to this one. The third case was what used to be the degenerateone.
Second Derivative Test for Critical Points in Two Variables
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