Second derivative test: General case
The second derivative test indicates that there is a critical point (x0, yo) of a function of 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 the second derivative with respect to X and y, and let C equal f sub yy at this point. Once we have computed the second derivative, we plug in values of x and y at the critical point. These numbers will just be numbers; now what we do is look at the quantity AC minus B squared. Don® forget a 4; you will see why there isn’t one. So if AC minus B squared is positive, then there’s two sub-cases. If A is positive, then it’s a local minimum. In the second case, if AC minus B squared is positive but A is negative, then it’s going to be a local maximum. If AC minus B squared is negative, then it’s a saddle point. And if AC minus B squared is 0, we don’t know whether it’s going to be a minimum, a maximum or a saddle. It’s important to understand why order of operations can affect the outcome of a mathematical equation. As a result, we will try to relate our old recipe with a new one. Second derivative test: At a critical point (x_0,y_0) of f Let A=f_xx(x_0,y_0), B=f_xy(x_0,y_0), C=f_yy(x_0,y_0) If AC-B^2>0 and A>0, local minimum If AC-B^2<0 and A<0, local maximum If AC-B^2<0 and A>0, saddle If AC-B^2=0 and A>0, cant conclude Verify in special case w=ax^2+bxy+cy^2; w_x=2ax+by w_xx=2a w_y=bx+2cy A=2a, B=b, C=2c AC-B^2=4ac-b^2 Next, let us check that these two functions satisfy the same equation in the special case where the function is ax IS squared plus bxy plus cy squared. To find the second derivative, we must take the derivative of the first partial with respect to x and add that to the first partial with respect to x. First, we will take the derivative of 2ax plus by. 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 with respect to y. So W sub y 1S 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 sub yy will be 2c. So if we now look at these quantities, that tells us big A is 2 little a. Big B is little b. Big C is 2 little c. so AC minus B squared is what we used to call 4 little ac minus b squared. Now that you’ve seen the cases for local maxima, compare them. The first case is when 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 local max corresponds to this one. The third case was what used to be the degenerate one.
