It is proved that f (x) = - x ^ 3 + 2 is a decreasing function on R by definition

It is proved that f (x) = - x ^ 3 + 2 is a decreasing function on R by definition

The definition domain of function f (x) = x & # 179; + 2 is (+ ∞, - ∞). If any two points x1, X2 are taken in the definition domain and X1 < X2, then
f(x1)－f(x2)＝(x1)³+2－[(x2)³-2]＝(x1)³－(x2)³＝[（x1)－(x2)] [(x1)²＋(x1)(x2)＋(x2)²]
∵ (x1) ＜ (x2), then (x1) - (x2) ＜ 0
Also ∵ (x1) &# 178; + (x1) (x2) + (x2) &# 178; = (x1) &# 178; + (x1) (x2) + &# 188; (x2) &# 178; + &# 190; (x2) &# 178; = [(x1) + (x2)] &# 178; + &# 190; (x2) &# 178;
Since (x1) ＜ (x2), that is, (x1) ≠ (x2), then [(x1) + (x2)] &# 178; > 0, &# 190; (x2) &# 178; ≥ 0
∴ [(x1)＋(x2)]²＋¾(x2)²＞0
∴f(x1)－f(x2)＝[（x1)－(x2)] [(x1)²＋(x1)(x2)＋(x2)²]＜0
Therefore, the function f (x) = x & # 179; + 2 is a decreasing function on R

If the monotone increasing interval of function f (x) = LG (x2 + 2x-3) is (a, + ∞), then a=______ ．

Let f (x) = LG (x2 + 2x-3) be a compound function, let x2 + 2x-3 be greater than 0, then the solution is x ＞ 1 or X ＜ - 3. Let f (x) = LGT, t = x2 + 2x-3. Because the outer function f (x) = LGT is an increasing function, the inner function T = x2 + 2x-3 is a decreasing function on (- ∞, - 3) and an increasing function on (1, + ∞)

Let function FX = x (e ^ x-1) - 1 / 2x ^ 2, then the monotone increasing interval of function FX is

FX = x (e ^ x-1) - 1 / 2x ^ 2F '(x) = e ^ X-1 + X * e ^ x-x = (1 + x) e ^ X - (1 + x) = (x + 1) (e ^ x-1) x + 1 is an increasing function, e ^ X-1 is an increasing function, so that (x + 1) (e ^ x-1) > = 0 ｛ x = 1 ｛ f (x) increasing interval is (- ∞, - 1] and [1, + ∞]. I'm glad to answer for you. I wish you progress in your study! If you don't understand, you can ask

FX = x (e ^ x-1) - 1 / 2x ^ 2, the monotone increasing interval of the function is
Given the function f (x) = x (e ^ x-1) - 1 / 2x ^ 2, find the monotone increasing interval of the function.

x=0

If f (x) = - x square + 2aX and G (x) = A / x + 2 are decreasing functions in the interval (1,5) (closed interval), then the value range of a is

The axis of symmetry of F (x) is a straight line x = a
∵ is a decreasing function in the interval [1,5]
∴a≤1
∵ g (x) is a decreasing function in the interval [1,5]
∴a>0
∴0

Find the maximum and minimum of function f (x) = XX − 1 in interval [2,5]

F ′ (x) = (x − 1) − x (x − 1) 2 = − 1 (x − 1) 2, when x ∈ [2,5], f ′ (x) < 0, so f (x) = XX − 1 is a decreasing function on [2,5], so the maximum value of F (x) is f (2) = 22 − 1 = 2, and the minimum value is f (5) = 55 − 1 = 54

If the difference between the maximum value and the minimum value of the function y = a ^ x (a > 0, and a ≠ 1) in the interval [1,2] is 1 / 4, then the real number a=___

Y = a ^ x (a > 0, and a ≠ 1) is a monotone function
The difference between the maximum value and the minimum value in the interval [1,2] is 1 / 4,
∴|f(2)-f(1)|=|a^2-a|=1/4
a^2-a=±1/4
（1） When a ^ 2-A = 1 / 4:
a^2-a-1/4=0
A = [1 + 1] / 2 = (1 + 2) / 2
A = (1-radical 2) / 2 < 0
A = (1 + radical 2) 2. (1)
（2） When a ^ 2-A = - 1 / 4:
a^2-a+1/4=0
(a-1/2)^2=0
a=1/2.(2)
A = 1 / 2, or a = (1 + radical 2) 2

For the inverse scale function y = - 2 / x, the following statement is true
A image passing through point (1,2)
B image is in the first three quadrants
C when x < 0, y increases with the increase of X
D image is an axisymmetric figure
I look at it and think that C and D are all right. What should I choose

For the inverse scale function y = - 2 / x, the following statement is true
C when x < 0, y increases with the increase of X
The image is centrosymmetric, not axisymmetric

As for the image of inverse scale function y = - 2x, the following statement is correct ()
A. When x is less than 0, the image is in the second quadrant D. the image is not axisymmetric

∵ k = - 2 ＜ 0, so the function image is located in two or four quadrants. In each quadrant, y increases with the increase of X. the image is axisymmetric, so a, B and D are wrong

Is y = 4 / X-1 an inverse scale function

Yes. Draw y = 4 / X on the XY plane,
Then move 1 unit to the right (in the direction of positive number x), that is y = 4 / (x-1)
If you move one unit down (in the direction of negative y), it is y = 4 / X - 1