一、从几道试题谈起1.一九八三年全国高考数学(文史类)试题五:在圆心为 O、半径为常数 R 的半圆板内画内接矩形(如图1),当矩形的长和宽各取多少时,矩形的面积最大?求出这个最大面积。2.一九八一年全国高考数学副卷(理王类)附加题;已知扇形 OAB 的中心角为45°,半径为R,矩形 PQMN 内接于这扇形,求矩形的对角线长1的最小值。(图2) First, from a few questions to talk about 1. 1983 mathematics (Cultural History class) test five questions: In the circle center O, radius R is a semi-circular plate inside the inscribed rectangle (Figure 1), when the rectangular When the length and width are taken as each other, what is the largest area of ​​the rectangle? Find the maximum area. 2. Supplementary title of the National College Entrance Examination Mathematics Supplement (King Wang Class) in 1981; the known sector OAB has a central angle of 45° and a radius of R. The rectangle PQMN is inscribed in this sector, and the diagonal of the rectangle is obtained. The minimum value of 1. (figure 2)