已知定义在R上的函数f(x)=(-2的x次方+1)/(2的x次方+a)是奇函数 (1)求a的值 (2)若对任意的t属于R,不等式f(t²-2t)+f(2t²-k)<0恒成立,求k的取值范围
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![已知定义在R上的函数f(x)=(-2的x次方+1)/(2的x次方+a)是奇函数 (1)求a的值 (2)若对任意的t属于R,不等式f(t²-2t)+f(2t²-k)<0恒成立,求k的取值范围](/uploads/image/z/5918059-19-9.jpg?t=%E5%B7%B2%E7%9F%A5%E5%AE%9A%E4%B9%89%E5%9C%A8R%E4%B8%8A%E7%9A%84%E5%87%BD%E6%95%B0f%28x%29%3D%28-2%E7%9A%84x%E6%AC%A1%E6%96%B9%2B1%29%2F%282%E7%9A%84x%E6%AC%A1%E6%96%B9%2Ba%29%E6%98%AF%E5%A5%87%E5%87%BD%E6%95%B0+%281%29%E6%B1%82a%E7%9A%84%E5%80%BC+%282%29%E8%8B%A5%E5%AF%B9%E4%BB%BB%E6%84%8F%E7%9A%84t%E5%B1%9E%E4%BA%8ER%2C%E4%B8%8D%E7%AD%89%E5%BC%8Ff%28t%26%23178%3B-2t%29%2Bf%282t%26%23178%3B-k%29%EF%BC%9C0%E6%81%92%E6%88%90%E7%AB%8B%2C%E6%B1%82k%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4)
已知定义在R上的函数f(x)=(-2的x次方+1)/(2的x次方+a)是奇函数 (1)求a的值 (2)若对任意的t属于R,不等式f(t²-2t)+f(2t²-k)<0恒成立,求k的取值范围
已知定义在R上的函数f(x)=(-2的x次方+1)/(2的x次方+a)是奇函数 (1)求a的值 (2)若对任意的t属于R,不等式f(t²-2t)+f(2t²-k)<0恒成立,求k的取值范围
已知定义在R上的函数f(x)=(-2的x次方+1)/(2的x次方+a)是奇函数 (1)求a的值 (2)若对任意的t属于R,不等式f(t²-2t)+f(2t²-k)<0恒成立,求k的取值范围
定义在R上的函数f(x)=(-2^x+1)/(2^x+a)是奇函数,则f(-x)=-f(x)
(-2^(-x)+1)/(2^(-x)+a) = -(-2^x+1)/(2^x+a)
-(2^(-x)+a) (-2^x+1) = (-2^(-x)+1)(2^x+a)
1+a*2^x-2^(-x)-a = -1-a*2^(-x)+2^x+a
(a-1)*2^x+(a-1)*2^(-x)+2(a-1) = 0
(a-1)*{2^x+2^(-x)+2 } = 0
2^x+2^(-x)+2 >0
∴a-1=0
∴a = 1
f(x) = (-2^x+1)/(2^x+1) = (-2^x-1+2)/(2^x+1) = -1 + 2/(2^x+1)
f(t²-2t)+f(2t²-k)<0
- 1 + 2/[2^(t^2-2t)+1] - 1 + 2/[2^(2t^2-k)+1] < 0
2/[2^(t^2-2t)+1] + 2/[2^(2t^2-k)+1] < 2
1/[2^(t^2-2t)+1] + 1/[2^(2t^2-k)+1] < 1
2^(t^2-2t)+1 >0,2^(2t^2-k)+1 >0
∴两边同乘以[2^(t^2-2t)+1 ] [2^(2t^2-k)+1]:
2^(2t^2-k)+1 + 2^(t^2-2t)+1 < [2^(t^2-2t)+1 ] [2^(2t^2-k)+1]
2^(2t^2-k) + 2^(t^2-2t) + 2 < 2^(2t^2-k) * 2^(t^2-2t) + 2^(2t^2-k) + 2^(t^2-2t) + 1
1 <2^(2t^2-k) * 2^(t^2-2t)
2^(2t^2-k+t^2-2t)>1
2^(3t^2-2t-k)>1
3t^2-2t-k>0
k>3t^2-2t=3(t^2-2/3t)=3(t-1/3)^2-1/3
又:3(t-1/3)^2≥0,即3(t-1/3)^2-1/3≥-1/3
∴k>-1/3