Is f (x) = - x + 2x + 1 x an increasing or decreasing function on [1, positive infinity]

Is f (x) = - x + 2x + 1 x an increasing or decreasing function on [1, positive infinity]

F (x) = - x ^ 2 + 2x + 1 is a decreasing function on [1, + infinity]. It is proved that any x2 > X1 > = 1F (x2) - f (x1) = (- x2 ^ 2 + 2x2 + 1) - (- X1 ^ 2 + 2x1 + 1) = - x2 ^ 2 + 2x2 + 1 + X1 ^ 2-2x1-1 = - (x2 ^ 2-x1 ^ 2) + 2 (x2-x1) = - (x2-x1) (x2 + x1) + 2 (x2-x1) = - (x2-x1) (x2 + x1-2) 0, X2 > 1, X1 > = 1, X2 + X1 > 2

Let f (x) be differentiable in a closed interval (1,2). It is proved that f (2) - f (1) = 3F '(x) / 2x, where 1 < x < 2

f(x)=(x^2 2x 1/2)/x
x∈[1,∝)
f`(x)=[(2x 2)x-(x^2 2x 1/2)]/x^2
=(2x^2 2x-x^2-2x-1/2)/x^2
=(x^2-1/2)/x^2
f`(x)

(100+101+102+…… +199）-（90+91+92+…… +189) simple calculation in the fourth grade of primary school

(100+101+102+…… +199）-（90+91+92+…… +189）
=（100-90）+（101-90）+…… +There were 100 groups (199-189)
=10×100
=1000

When a cylinder is cut into two cylinders with a height of 4cm and 6cm respectively, the surface area is increased by 12.56cm2, and the original volume of the cylinder is []
When a cylinder is cut into two cylinders with a height of 4cm and 6cm respectively, the surface area is increased by 12.56cm2. The original volume of the cylinder is [must be given before 3.22cm]

The increased surface area is 2 bottom areas
Bottom area = 12.56/2 = 6.28 square centimeter
The original cylinder volume = 6.28x (6 + 4) = 62.8 cubic centimeter

The solution equation: 5.4 + x = 9.26 x △ 4 = 14.5

x=9.26-5.4
x=3.86
X = 14.5 times 4
x=58

As shown in the figure: trapezoid area is 45 square meters, height is 6 meters, triangle AED area is 5 square meters, calculate the shadow area

S △ Abe = area of trapezoid - s △ bcd-s △ AED,
=45-10×6÷2-5,
=10 square meters;
S shadow = s △ abc-s △ Abe,
=10×6÷2-10,
=20 square meters;
A: the shadow area is 20 square meters;

How to calculate 12.5 times 0.25 times 3.2

12.5*0.8*0.25*4
=10*1=10
（3.2=0.8*4）

In the triangle ABC, if ad is the height of edge BC and ad = BC, the maximum value of B / C + C / b

From the area relation, we can get a & # 178; = bcsina ①
From the cosine theorem, we get a & # 178; = B & # 178; + C & # 178; - 2bccosa ②
Substituting ① into ②, BC (Sina + 2cosa) = B & # 178; + C & # 178;
That is, B / C + C / b = B & # 178; + C & # 178; / BC = Sina + 2cosa = √ 5sin (a + α) ≤ √ 5
Where Tan α = 2
Therefore, the maximum value of B / C + C / B is √ 5
We can also calculate sin α = 2 / √ 5
So the value range is: (2, √ 5]
The reason of sina + 2cosa = √ 5sin (a + α)
It is derived from the following formula
Asinα+Bcosα
= √（A+B）[A/√（A+B)* sinα+ B/√（A+B）cosα]
=√（A+B） sin(α+φ)
Where sin φ = B / √ (a + b), cos φ = A / √ (a + b)
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Given a, B, C ∈ (0, + ∞) and a + B + C = 1, we prove that: (1a-1) (1b-1) (1c-1) ≥ 8

It is proved that: ∵ a, B, C ∈ (0, + ∞) and a + B + C = 1, ∵ (1a-1) (1b-1) (1c-1) = (1 − a) (1 − b) (1 − C) ABC = (B + C) (a + C) (a + b) ABC ≥ 2BC · 2Ac · 2ababc = 8. If and only if a = b = C = 13, the equal sign holds

The area of a trapezoid is 240 square decimeters, the upper bottom is 12 decimeters, and the lower bottom is 18 decimeters. How high is the trapezoid?

240x2 △ 12 + 18 = 480 △ 30 = 16 (decimeter)