高中数学，椭圆以及函数求导 20
(1)
M(0, 2), b = 2
e² = c²/a² = (a² - b²)/a² = 1 - b²/a² = 1 - 4/a² = 2/3, a² = 12
椭圆方程: x²/12 + y²/4 = 1
(2)
直线: y = k(x - 2)
代入椭圆方程并整理: (3k² + 1)x² - 12k²x + 12(k² - 1) = 0x₁ + x₂ = 12k²/(3k² + 1)
x₁x₂ = 12(k² - 1)/(3k² + 1)
A(x₁, y₁), B(x₂, y₂)
t = 向量OA•向量OB = x₁x₂ + y₁y₂ = x₁x₂ + k(x₁ - 2)*k(x₂ - 2)
= (k² + 1)x₁x₂ - 2k²(x₁ + x₂) +4k²
= (k² + 1)*12(k² - 1)/(3k² + 1) - 2k²*12k²/(3k² + 1) +4k²
= 4(k² - 3)/(3k² + 1)
要使为AOB锐角，只须t > 0, k² - 3 > 0, k > √3 或k < -√3
(1)
a = 0, f(x) = 2x - lnx
f'(x) = 2 - 1/x = 0, x = 1/2
0 < x < 1/2: f'(x) < 0
x > 1/2: f'(x) > 0
极小值f(1/2) = 2*(1/2) - ln(1/2) = 1 + ln2
(2)
由(1)可知，显然a = 0不满足要求
f'(x) = ax + 2 - 1/x = (ax² + 2x - 1)/x = 0
ax² + 2x - 1 = 0x = [-1 ±√(a+ 1)]/a
下面分情况况讨论
(i) a > 0
ax² + 2x - 1为开口向上的抛物线，要使f(x)在[1/3, 2]上为增函数，只需[-1 -√(a+ 1)]/a ≥ 2或[-1 +√(a+ 1)]/a ≤ 1/3
[-1 -√(a+ 1)]/a ≥ 2可变为-√(a+ 1) ≥ 2a + 1 >0，显然不可能
[-1 +√(a+ 1)]/a ≤ 1/3, √(a+ 1) ≤ a/3 + 1平方并整理: a(a - 3) ≥ 0
a ≥ 3 (舍去a ≤ 0)
(ii) a < 0
ax² + 2x - 1为开口向下的抛物线，要使f(x)在[1/3, 2]上为增函数，只需[-1 +√(a+ 1)]/a ≥ 2且[-1 -√(a+ 1)]/a ≤ 1/3
与(i)类似，从[-1 +√(a+ 1)]/a ≥ 2可得a ≤ -3/4
从[-1 -√(a+ 1)]/a ≤ 1/3可得a ≤ 0二者结合: a ≤ -3/4
(i)(ii)结合: a ≤ -3/4或a ≥ 3
