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Math 210 Jerry L. Kazdan Quadratic Polynomials Polynomials in One Variable. After studying linear functions y = ax + b , the next step is to study quadratic poly- nomials , y = ax 2 + bx + c , whose graphs are parabolas. Initially one studies the simpler special case y = ax 2 + c (1) If a > 0 these parabolas have a minimum at x = 0 and open upward, while if a < 0 they have a maximum at x = 0 and open downward. One can reduce the more general quadratic polynomial y = ax 2 + bx + c (2) to the special case (1) by a change of variable, x = v + r translating x by r , where r is to be found. Substituting this into (2) we find y = a ( v + r ) 2 + b ( v + r ) + c = av 2 + ( b + 2 ar ) v + ar 2 + br + c. We now pick r to remove the linear term in v , that is, b + 2 ar = 0 so r = b/ (2 a ). Then y = av 2 + k = a ( x r ) 2 + k, (3) where k = c b 2 / (4 a ). Thus, x = r is the axis of symmetry of this parabola. This procedure is equivalent to “completing the square”, a procedure that should be familiar from algebra. Another way to find r is to observe that the only critical point of (2) is at x = b/ (2 a ). Thus translating by b/ (2 a ) places this critical point on the vertical axis. 0 5 10 15 20 25 -4 -2 2 4 6 Shifted Parabola Example . On the right are the graphs of y = x 2 + 1 and y = ( x 2) 2 + 1 = x 2 2 x + 2. They clearly shows the graph on the right is merely a translation of the graph on the left. Polynomials in Several Variables. Maxima, minima, and saddle points. There are more interesting possibilities for quadratic polynomials in two variables. w = 2 x 2 + y 2 w = (2 x 2 + y 2 ) w = 2 x 2 + y 2 From the graphs it is clear that the first has a minimum at the origin, the second a maxi- mum , while the third has a saddle point there. 1

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It is less obvious how to treat polynomials such as w = 3 x 2 2 xy + y 2 (4) or w = 3 x 2 2 xy + y 2 + z 2 (5) with xy terms and possibly more variables. These two examples can be handled if one recognizes that x 2 2 xy + y 2 = ( x y ) 2 so, if one makes the change of variables r = x, s = x y and t = z then in the new coordinates these polynomials become w = 2 x 2 + ( x y ) 2 = 2 r 2 + s 2 and w = 2 r 2 + s 2 + t 2 , which clearly have minima at the origin in rs and rst space, respectively. The primary task of this section is to give useful criteria for a quadratic polynomial in several variables to have a maximum, minimum, or saddle point. This will then be used in the next section to generalize the calculus of one variable second derivative test for a local maximum to functions of several variables such as our quadratic polynomials. The first step is to be a bit more systematic. Rewrite (4) as w = 3 x 2 xy yx + y 2 and observe that using the inner (=dot) product it can be written in the more compact form w = X · A X , (6) where X = parenleftBigg x y parenrightBigg and A is the symmetric matrix A = parenleftBigg 3 1 1 1 parenrightBigg .
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