題目
4. Let
f(x,y)=x2+4xy+y2. We call the region
{(x,y)∣x2+y2<1} open disk
D1,
{(x,y)∣x2+y2≤1} the closed disk
D2, and
{(x,y)∣x2+y2=1} the circle
C.
i) (10%) On the open disk D1 find, if there is any, those points on which f(x,y) is a relative minimum, relative maximum or a saddle point.
ii) (10%) On the circle C compute the absolute maximum value and absolute minimum value of f(x,y).
iii) (10%) On the closed disk D2 give your reasons to find the absolute maximum value and absolute minimum value of f(x,y).
解答
解法一
思路
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- 先用 Hessian 判別開圓盤的唯一臨界點;圓周上改用 x2+y2=1,再與內部比較閉圓盤的全域極值。
答題過程
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梯度為 (2x+4y,4x+2y),唯一臨界點是 (0,0)。Hessian 行列式 4−16=−12<0,故
(0,0) 是鞍點.
圓周 C 上,
f=x2+y2+4xy=1+4xy.
由 −1/2≤xy≤1/2,圓周最大、最小值為
3(於 x=y=±1/2),−1(於 x=−y=±1/2).
因 f 是二次齊次函數,沿每條由原點出發的射線只會乘上非負的平方因子;故閉圓盤的絕對最大、最小值仍分別是 3、−1。