趣味數學問題懸賞3 初三，高一組參考答案

2022-11-02 20:53 作者:数学只因 0人读过 | 我要投稿

hihi

第一次寫專欄，可能寫的不是太好

不知道啥是趣味數學問題懸賞看這裏

第一題解答：

(1)

相信我們都學過這件事：對於一般二次函數 $y%3Dax%5E2%2Bbx%2Bc$ ，其頂點坐標為 $(-%5Cfrac%7Bb%7D%7B2a%7D%20%2C%20%5Cfrac%7B4ac-b%5E2%7D%7B4a%7D%20)$ 。把 $a%3D1%20%2C%20b%3D-(k-1)%2C%20c%20%3D%20-k-1$ 帶入公式，得頂點坐標為 $%EF%BC%88%5Cfrac%7Bk-1%7D%7B2%7D%2C%5Cfrac%7B-k%5E2-2k-5%7D%7B4%7D%EF%BC%89$

(2)

$%5Ctriangle%7BABC%7D%E7%9A%84%E9%9D%A2%E7%A9%8D%3D%5Cfrac%7B(A%E5%92%8CB%E4%B9%8B%E9%96%93%E7%9A%84%E8%B7%9D%E9%9B%A2)%5Ccdot(%E9%BB%9EC%E5%88%B0AB%E7%9A%84%E5%9E%82%E7%9B%B4%E8%B7%9D%E9%9B%A2)%7D%7B2%7D%5C%5C%3D%5Cfrac%7B(%E4%BA%8C%E6%AC%A1%E6%96%B9%E7%A8%8Bx%5E2-(k-1)x-k-1%3D0%E5%85%A9%E6%A0%B9%E7%9A%84%E8%B7%9D%E9%9B%A2)%5Ccdot(%E9%BB%9EC%E7%9A%84y%E5%9D%90%E6%A8%99%E7%9A%84%E7%B5%95%E5%B0%8D%E5%80%BC)%7D%7B2%7D%5C%5C%3D%5Cfrac%7B(%5Csqrt%7Bk%5E2%2B2k%2B5%7D)%5Ccdot(%5Cfrac%7Bk%5E2%2B2k%2B5%7D%7B4%7D)%7D%7B2%7D%5C%5C%3D%5Cfrac%7B(%5Csqrt%7Bk%5E2%2B2k%2B5%7D)%5E3%7D%7B8%7D%5C%5C%3D%5Cfrac%7B(%5Csqrt%7B(k%2B1)%5E2%2B4%7D)%5E3%7D%7B8%7D%5C%5C%5Cgeq%5Cfrac%7B(%5Csqrt%7B4%7D)%5E3%7D%7B8%7D%3D1$

第二題解答：

$%E5%AE%9A%E7%BE%A9%20f(x)%3D%5Csqrt%7B3-x%7D-%5Csqrt%7Bx%2B1%7D%EF%BC%8Cs_%7B1%7D%20(x)%3D%5Csqrt%7B3-x%7D%2C%20s_%7B2%7D(x)%3D-%5Csqrt%7Bx%2B1%7D%5C%5C%0A%E6%B3%A8%E6%84%8F%E5%88%B0%20%5Cforall%20x%5Cin%20%5B-1%2C3%5D%2Cf(x)%3Ds_%7B1%7D(x)%2Bs_%7B2%7D(x)%5C%5C%0A%5Cbecause%20s_%7B1%7D(x)%E5%92%8Cs_%7B2%7D(x)%E5%9C%A8%5B-1%EF%BC%8C3%5D%E7%9A%84%E5%8D%80%E9%96%93%E7%82%BA%E5%9A%B4%E6%A0%BC%E9%81%9E%E6%B8%9B%E3%80%82%0A%5Ctherefore%20f(x)%E5%9C%A8%5B-1%EF%BC%8C3%5D%E7%9A%84%E5%8D%80%E9%96%93%E7%82%BA%E5%9A%B4%E6%A0%BC%E9%81%9E%E6%B8%9B%E3%80%82%5C%5C%0A%E6%B3%A8%E6%84%8F%E5%88%B0f(x)%E5%8F%AA%E5%9C%A8%5B-1%2C3%5D%E5%8D%80%E9%96%93%E5%86%85%E6%89%8D%E6%9C%89%E6%84%8F%E7%BE%A9%5C%5C%0A%5Ctherefore%20f(x)%E7%82%BA%E5%9A%B4%E6%A0%BC%E9%81%9E%E6%B8%9B%5C%5C%0A%E6%B3%A8%E6%84%8F%E5%88%B0%20f(%5Cfrac%7B8-%5Csqrt%7B31%7D%7D%7B8%7D)%3D%5Cfrac%7B1%7D%7B2%7D%5C%5C%0A%5Ctherefore%5Cforall%20x%5Cin%20(-%5Cinfty%2C%5Cfrac%7B8-%5Csqrt%7B31%7D%7D%7B8%7D)%5Ccap%20%5B-1%2C3%5D%2C%20f(x)%3Ef(%5Cfrac%7B8-%5Csqrt%7B31%7D%7D%7B8%7D)%3D%5Cfrac%7B1%7D%7B2%7D%5C%5C%0A%5Ctherefore%5Cforall%20x%5Cin%20%5B-1%2C%5Cfrac%7B8-%5Csqrt%7B31%7D%7D%7B8%7D)%2C%20f(x)%3E%5Cfrac%7B1%7D%7B2%7D%0A$