Proving inequality with mean value theorem X > 3-1 / X (x > 0)

Proving inequality with mean value theorem X > 3-1 / X (x > 0)

2 √ x + 1 / x = √ x + √ x + 1 / X ≥ 3 * 1 = 3 under the root of 3 times
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Proving inequality | arctanx arctany|

Let f (a) = arctan (a), f '(a) = 1 / (1 + A & # 178;)
F (a) is continuously differentiable in (x, y),
| arctanx - arctany | = 1/(1 + c²) * | x - y | < | x - y |,c∈(x,y)
When a = b = 0, arctanx = arctany = 0
∴
| arctanx - arctany | ≤ | x - y |

The minimum value of Xe to the x power

y=xe^x
y'=e^x+xe^x=(1+x)e^x=0
x=-1
When x = - 1, there is a minimum y = - 1 / E