It is proved that the function y = 1 / x + 1 is a decreasing function at (- 1. + ∞)

It is proved that the function y = 1 / x + 1 is a decreasing function at (- 1. + ∞)

Let a > b > - 1
Then f (a) - f (b) = 1 / (a + 1) - 1 / (B + 1) = [(B + 1) - (a + 1)] [(a + 1) (B + 1)]
=(b-a)/[(a+1)(b+1)]
a> B, so b-a-1, b > - 1
So a + 1 > 0, B + 1 > 0
So when (B-A) / [(a + 1) (B + 1)] b > - 1
f(a)

It is proved that the function y = - x ^ 2 + 2x is a decreasing function on (1, positive infinity)
Such as the title, a function of high school

There are many ways to prove this problem
The simplest method is formulation
y=-(x-1)2+1
So it's a decreasing function on (1, positive infinity)
The second is to prove by definition
I won't go into details here
Don't know how to hi me again

It is proved that the function y = x ^ 2 + 2x-3 is a decreasing function on (- ∞, 1)

It should be: (- ∞, - 1) is a decreasing function
prove:
y=x^2+2x-3
Let x1

(1)(m+n)^-4m(m+n)+4m^
(2)a^+2a(b+c)+(b+c)^
(3)4+12(x-y)+9(x-y)^
(4)(a-b)(x-y)-(b-a)(x+y)

1.[（m+n)-2m]
2.[a+(b+c)]
3.[2+（x-y)]
4.(a-b)(x-y+x+y)
=2x(a-b)

Quadratic function y = 1 / 2x ^ 2 + 3x + 5 / 2 problem solving process

y=1/2（x^2+6x+5） y=1/2（x+1）（x+5）

Parabola y = 2x square 3x-5, passing through point a (n, 9), finding the value of n

If the parabola y = 2x & # 178; - 3x-5 passes through point a (n, 9), then x = n, y = 9 is substituted into the parabola
9=2n²－3n－5
2n²－3n－14=0
(2n－7)(n＋2)=0
We get: n = 7 / 2 or n = - 2

It is known that the parabola y = x & # 178; + BX + C intersects the X axis at point a (α, 0) and point B (β, 0), and the images of y = - X-2 and y = - 3 / X all pass through point m (α, β)
Ask for the value of B and C!

Y = - X-2 and y = - 3 / X eliminate y simultaneously
-x-2=-3/x
x²+2x-3=0
The solution is X1 = - 3, X2 = 1
∴y1=1,y2=-3
∴α=-3,β=1
∵ parabola y = x & # 178; + BX + C intersects with X axis at points a (α, 0) and B (β, 0)
The two roots of X & # 178; + BX + C = 0 are α = - 3, β = 1
According to Weida's theorem:
-b=α+β=-2,c=αβ=-3
∴b=2,c=-3

A craft shop can make a profit of 45 yuan if it sells a certain craft at the marked price. If 12 pieces of the craft are sold at a 25% discount on the marked price, the profit is equal. How much is the purchase price of each piece? (urgent!)

(1) Suppose the purchase price of each handicraft is x yuan,
The price is (x + 45) yuan,
According to the meaning of the title, we can draw a conclusion
8×[85%·(x+45)-x]=12×(45-35)
The solution is x = 155, x + 45 = 200
Therefore, the purchase price of each handicraft is 155 yuan and the price is 200 yuan

Find the trajectory equation with the square difference of the distance between O (0,0). A (c.0) as constant C

Let P coordinate be (x, y)
PO=√(x^2+y^2)
PA=√[(x-c)^2+y^2]
|P0^2-PA^2|=|x^2+y^2-[(x-c)^2+y^2]|
=|x^2-(x-c)^2|=C
Two sides square
x^2-(x-C)^2=C^2
x^2-(x^2-2CX+C^2)=C^2
x^2-x^2+2Cx-C^2=C^2
2CX=2C^2
x=C

Given that a is a root of the equation x ^ 2-5x + 1 = 0, find the value of a ^ 2 + A ^ 2

Because the product of two is 1, the other B is 1 / A
a+b=5,ab=1
a^2+1/a^2=a^2+b^2=(a+b)^2-2ab=25-2=23