Function f (x) = inx-a (x-1) / X (x > 0, a belongs to R) (1) Try to find the monotone interval of F (x) (2) When a > 0, it is proved that the necessary and sufficient condition for the image of function f (x) to have a unique zero is a = 1

Function f (x) = inx-a (x-1) / X (x > 0, a belongs to R) (1) Try to find the monotone interval of F (x) (2) When a > 0, it is proved that the necessary and sufficient condition for the image of function f (x) to have a unique zero is a = 1

（1）f'(x)=1/x-a/x^2=(x-a)/x^2 (x>0)
When A0 increases
When 00 increases
When 0A, f '(x) > 0 increases
When (00)
It is easy to know that when a = 1, f (a) = f (1) = 0
Let's prove that lna-a + 1 = 0 has a unique root a = 1
f'(a)=1/a-1=(1-a)/a
Then when a = 1, f '(a) = 0
When 00 increases
Therefore, when a = 1, the minimum value of F (a) is f (1) = 0
Therefore, if and only if a = 1, there is f (a) = 0
That is, lna-a + 1 = 0 has a unique root a = 1
To sum up, the necessary and sufficient condition for the image of function f (x) to have a unique zero is a = 1
Finally finished!

The known function f (x) = x + A / X (a belongs to R), G (x) = INX 1. Find the monotone interval of function f (x) = f (x) + G (x) 2. If the equation g (x) / x ^ 2 = f (x) - 2e (E is the base of natural number) about X has only one real root, find the value of A Some conversions are best explained simply why,)

∵F(X)=f(X)+g(X)=X+a/X+InX(X>0)
∴F'(X)=1-a/X^2+1/X
Let f '(x) = 0,1 / x = t (T > 0)
Then - at ^ 2 + T + 1 = 0, T1 = [- 1 - √ (1 + 4a)] / (- 2A), T2 = [- 1 + √ (1 + 4a)] / (- 2A)
① If 1 + 4A ≤ 0, a ≠ 0, a ≤ - 1 / 4, then f '(x) ≤ 0, f (x) monotonically decreases at (0, + ∞)
② If 1 + 4A ≥ 0 and a ≠ 0, a ≥ - 1 / 4, suppose T1 ≤ 0 and T2 ≥ 0, that is [- 1 - √ (1 + 4a)] / (- 2A) ≤ 0 and [- 1 + √ (1 + 4a)] / (- 2A) ≥ 0
When a ∈ (- 1 / 4,0), T1 ≤ 0 is always true, [- 1 + √ (1 + 4a)] / (- 2A) ≥ 0 = > 1 + 4A ≥ 1 = > a ≥ 0 is not true
When a ∈ (0, + ∞), T1 ≤ 0 does not hold, and [- 1 + √ (1 + 4a)] / (- 2A) ≥ 0 = > a ≤ 0 does not hold
When a ∈ (- 1 / 4,0), T1, T2 ≤ 0; When a ∈ (0, + ∞), T1 > 0, T2

If the function f (x ^ n) = INX is known, the value of F2 is

Let x ^ n = t, then x = T ^ (1 / N)
So f (x ^ n) = INX can be reduced to
f(t)=int^(1/n)=lnt/n
That's f (x) = LNX / n
So f (2) = LN2 / n

Find the maximum value of F (x) with known function f (x) = inx-x-1

f'(x)=1/x-1=(1-x)/x
The domain is x > 0
Then 0

Known function f (x) = inx-1 / 2aX ^ 2-2x (a)

1) F ′ (x) = 1 / X - a X-2. If f (x) has a monotonic decreasing interval, then f ′ (x) ≤ 0 on (0, + ∞), a ≥ 1 / X ²- 2/x=(1/x -1) ²- 1 ≥ - 1, that is, a ∈ [- 1 + ∞) 2) if a = - 1 / 2, f (x) = - 1 / 2 x + B can be transformed into LNX + 1 / 4 x ^ 2-3 / 2 x = B, so that G (x) = LNX + 1 / 4 x ^ 2-3 / 2 x

Monotone interval of function f (x) = 2x2 INX

Domain x > 0
f'(x)=4x-1/x
F '(x) > 0 = > x > 1 / 2 monotonic increase
f'(x) 0