定义函数f(x)=max(x^2,x^-2)x∈(-∞,0)∪(0,+∞)求f(x)的最小值
来源:学生作业帮助网 编辑:作业帮 时间:2024/06/16 03:20:08
![定义函数f(x)=max(x^2,x^-2)x∈(-∞,0)∪(0,+∞)求f(x)的最小值](/uploads/image/z/1001684-20-4.jpg?t=%E5%AE%9A%E4%B9%89%E5%87%BD%E6%95%B0f%28x%29%3Dmax%28x%5E2%2Cx%5E-2%29x%E2%88%88%28-%E2%88%9E%2C0%29%E2%88%AA%280%2C%2B%E2%88%9E%29%E6%B1%82f%28x%29%E7%9A%84%E6%9C%80%E5%B0%8F%E5%80%BC)
定义函数f(x)=max(x^2,x^-2)x∈(-∞,0)∪(0,+∞)求f(x)的最小值
定义函数f(x)=max(x^2,x^-2)x∈(-∞,0)∪(0,+∞)求f(x)的最小值
定义函数f(x)=max(x^2,x^-2)x∈(-∞,0)∪(0,+∞)求f(x)的最小值
令x^2≥x^-2,化简
x^2-x^-2≥0
x^4-1≥0
(x^2-1)(x^2+1)≥0
x^2-1≥0
(x-1)(x+1)≥0
x≤-1或x≥1
也就是说,当x≤-1或x≥1时,x^2≥x^-2,所以f(x)=max(x^2,x^-2)=x^2
同理可得,当-1≤x
很明显这个最小值是当x^2=x^-2时出现,即x=±1时出现,最小值是1
令x^2≥x^-2,化简
x^2-x^-2≥0
x^4-1≥0
(x^2-1)(x^2+1)≥0
x^2-1≥0
(x-1)(x+1)≥0
x≤-1或x≥1
也就是说,当x≤-1或x≥1时,x^2≥x^-2,所以f(x)=max(x^2,x^-2)=x^2
同理可得,当-1≤x<0或0
全部展开
令x^2≥x^-2,化简
x^2-x^-2≥0
x^4-1≥0
(x^2-1)(x^2+1)≥0
x^2-1≥0
(x-1)(x+1)≥0
x≤-1或x≥1
也就是说,当x≤-1或x≥1时,x^2≥x^-2,所以f(x)=max(x^2,x^-2)=x^2
同理可得,当-1≤x<0或0
收起
令x^2≥x^-2,化简
x^2-x^-2≥0
x^4-1≥0
(x^2-1)(x^2+1)≥0
x^2-1≥0
(x-1)(x+1)≥0
x≤-1或x≥1
也就是说,当x≤-1或x≥1时,x^2≥x^-2,所以f(x)=max(x^2,x^-2)=x^2
同理可得,当-1≤x<0或0
全部展开
令x^2≥x^-2,化简
x^2-x^-2≥0
x^4-1≥0
(x^2-1)(x^2+1)≥0
x^2-1≥0
(x-1)(x+1)≥0
x≤-1或x≥1
也就是说,当x≤-1或x≥1时,x^2≥x^-2,所以f(x)=max(x^2,x^-2)=x^2
同理可得,当-1≤x<0或0
收起