Let the function f (x) be second-order differentiable on the interval [0,1], and f (0) = 0, f '' (x) > 0. It is proved that f (x) / X is a monotonically increasing function on (0,1]

Because f '' (x) > 0
So f '(x) is an increasing function
If f (0) = 0, then f '(x) monotonically increases in (0,1] and f' (x) > 0
So the proposition is proved

The function f (x) is continuous on the interval [0,2a], and f (0) = f (2a), which is proved; There is at least one point on [0, a] such that f (x) = f (x + a)

Let f (x) = f (x) - f (x + a)
Then f (0) = f (0) - f (a)
F（a）=f（a）-f（2a）=f（a）-f（0）
So f (0) × F (a) less than 0
According to the zero point theorem, there is e so that f (E) = 0 is the result

It is proved that the quadratic function f (x) = ax ＾ 2 + BX + C (a < 0) is an increasing function on the interval (- ∞, - B / 2a)

- B / 2a is the coordinate of the fixed point X of the function,
A is less than 0, so the function is a parabola with an opening downward and has a maximum at x = - B / 2A
Therefore, f (x) = ax ＾ 2 + BX + C (a < 0) is an increasing function on the interval (- ∞, - B / 2a)

Known function f (x) = (x + 1) LNX X − 1 (x > 0 and X ≠ 1) (1) The monotonicity of function f (x) is discussed (2) Proof: F (x) > 2

(1) ∵ f (x) = (x + 1) lnxx − 1 (x > 0 and X ≠ 1) ∵ f ′ (x) = − 2lnx + X − 1x (x − 1) 2 let g (x) = − 2lnx + X − 1x then G ′ (x) = − 2x + 1 + 1x2 = (x − 1x) 2 is constant from G ′ (x) ≥ 0, G (x) monotonically increases at (0, + ∞), and ∵ g (1) = 0, so when x ∈ (0, 1), G

Given the function f (x) = (LNX + a) / X (a ∈ R), when a = 1 and X ≥ 1, it is proved that f (x) ≤ 1

Just prove that LNX + 1 ≤ X
Let g (x) = LNX - x + 1
g'(x)=1/x-1
When x > = 1, G '(x)

It is proved that the function f (x) = lnx-x2 + X has only one zero

It is proved that: F (x) = lnx-x2 + X, its definition field is (0, + ∞),  f ′ (x) = 1x − 2x + 1 = − 2x2 − x − 1X, so that f '(x) = 0, that is − 2x2 − x − 1x = 0, the solution is x = − 12 or x = 1. ? x > 0, Ƚ x = − 12 is rounded off. When 0 < x < 1, f' (x) > 0; When x > 1, f '(x) < 0

Let the function f (x) be second-order differentiable on the interval [0,1], and f (0) = 0, f '' (x) > 0. It is proved that f (x) / X is a monotonically increasing function on (0,1]