Advanced High-School Mathematics

SECTION 3.2 Classical Inequalities 151 Proof. We define a quadratic function ofx: Q(x) = (xx 1 −y 1 )^2 +···(xxn−yn)^2 = (x^21 + ...

152 CHAPTER 3 Inequalities Find the maximum value of the objective functionx+y+zgiven thatx^2 +y^2 +z^2 = 4. (Hint: use AM(x,y, ...

SECTION 3.2 Classical Inequalities 153 (iv) Show that this implies thatP(x,y)≥0 whenx, y >0 with equality if and only ifx=y.) ...

154 CHAPTER 3 Inequalities x y y=x n. . . . Q=Q(b,b ) n P=P(a,a ) n (S (a,b),0)n Equation 1: y=x² Show that ifa, b >0, then ...

SECTION 3.3 Jensen’s Inequality 155 the objective is to show thatP(a,b)≥0 whena,b,m,nare as given above. Regard the above as a p ...

156 CHAPTER 3 Inequalities 10 8 6 4 2 -2 -4 -6 -8 5 10 P 5 P 6 P 3 P 4 P 2 P 1 Next, recall from elementary differential calculu ...

SECTION 3.4 Holder Inequality ̈ 157 true forn= 2. We may assume that 0≤tn<1; set x 0 = t 1 x 1 +···+tn− 1 xn− 1 1 −tn ; note ...

158 CHAPTER 3 Inequalities y = ln x x y P Q Proof. The proof involves a ge- ometrical fact about the graph of the function y = l ...

SECTION 3.4 Holder Inequality ̈ 159 We may assume that both A, B 6 = 0, else the theorem is clearly true. Therefore by using You ...

160 CHAPTER 3 Inequalities Indeed, from the above, we have by settingr= q p >1 that ∑n i=1 xip/n = AM(xp 1 ,xp 2 ,...,xpn) ≤ ...

SECTION 3.5 Quadratic Discriminant 161 3.5 The Discriminant of a Quadratic The discriminant of a quadratic polynomial, while fin ...

162 CHAPTER 3 Inequalities The D = 0 case is the one we shall find to have many applica- tions, especially to constrained ex- tr ...

SECTION 3.5 Quadratic Discriminant 163 Solving for the discriminant in terms ofm, we quickly obtain 0 = b^2 − 4 ac = (−12)^2 − 4 ...

164 CHAPTER 3 Inequalities We have 0 = b^2 = 4ac = m^2 − 4 which immediately givesm =±2. Only the valuem = 2 is rele- vant here ...

SECTION 3.5 Quadratic Discriminant 165 Example 4. Given that x^2 + 2y^2 = 6, find the maximum value of x+y. Solution. This probl ...

166 CHAPTER 3 Inequalities Therefore, the maximum value of x+y is 3 (and the minimum value ofx+yis−3). Exercises. Given thatx+y ...

SECTION 3.6 Cubic Discriminant 167 x y P(a,0) R Q(b,0) x + y = r 2 22 Equation 1: x²+y²=6 Above is depicted the circle whose eq ...

168 CHAPTER 3 Inequalities problems. This raises at least a couple of questions. The immediate question raised here would be whe ...

SECTION 3.6 Cubic Discriminant 169 Case (i): ∆>0. That is to say, (x 2 −x 1 )^2 >0 and so certainly the zerosx 1 andx 2 ar ...

170 CHAPTER 3 Inequalities With a bit of effort, this determinant can be expanded. It’s easier to first compute the determinant ...