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在约束条件Ａｘ2 +Ｂｘｙ +Ｃｙ2 =Ｍ下 ,求函数ω =Ａｘ2 +Ｄｘｙ+Ｃｙ2 (Ａ、Ｃ、Ｍ∈Ｒ+,Ｂ、Ｄ∈Ｒ)的最值 ,可采用解几或三角换元的方法求解 ,但计算繁杂 .若抓住条件式和欲求式中ｘ2 、ｙ2 系数相同这个特征 ,利用不等式 (ａ±ｂ) 2 ≥ 0 ,可简易地解决这类问题 .例 1
Under the constraint Ax2 + Bxy + Cy2 = M, find the maximum value of the function ω = Ax2 + Dxy + Cy2 (A, C, M ∈ R +, B, D ∈ R). The method is solved, but the calculation is complicated. If we take advantage of the fact that the coefficients of x2 and y2 are the same in the conditional expression and the desire expression, the inequality (a±b) 2 ≥ 0 can easily solve the problem. Example 1