The function f (x) = x + 1 / X proves that f (x) is a decreasing function on (0,1)

The function f (x) = x + 1 / X proves that f (x) is a decreasing function on (0,1)

prove:
Any 0

It is proved that the function f (x) = x2-2 is an increasing function on (0, + OO)
X2 (square of x)
+OO (positive infinity)

Let a > b > 0
∴ f(a)=a²-2,f(b)=b²-2
∴f(a)-f(b)=a²-b²=(a-b)(a+b)
∵a＞b＞0
∴a-b＞0,a+b＞0
∴(a-b)(a+b)=f(a)-f(b)＞0
∴f(a)＞f(b)
Ψ f (x) = x & sup2; - 2 is an increasing function between (0, + ∞)

Try to prove that the function y = x + 1 / X is an increasing function on (1, + OO)

y'=1-1/x^2=(x^2-1)/x^2
X is on (1, + OO)
X ^ 2 > 0 x ^ 2-1 > 0 so y '> 0
So the function y = x + 1 / X is an increasing function on (1, + OO)