Let f (x) = (SiNx) ^ 2 + (cosx) ^ 2, and use the differential mean value theorem to prove that for all x belonging to negative infinity to positive infinity, there is always f (x) = 1

Let f (x) = (SiNx) ^ 2 + (cosx) ^ 2, and use the differential mean value theorem to prove that for all x belonging to negative infinity to positive infinity, there is always f (x) = 1

For all real numbers x,
Because f '(x) =. = 0,
So f (x) = constant C,
Take x = 0 and get C = 1,
So f (x) is constant = 1

Prove that the equation x ^ 2 + x = 1 has only one positive root (differential mean value theorem)

Let f (x) = x ^ 2 + X-1 (x > 0)
Then f '(x) = 2x + 1 > 0 (x > 0), so f (x) rises monotonically in [0, + ∞)
And f (0) = - 10
According to the existence theorem of zero point of continuous function, there exists x0 ∈ (0,1), f (x0) = 0
Since f (x) rises monotonically in [0, + ∞), there is a unique positive root x0 of F (x) in [0, + ∞)

Using the differential mean value theorem to prove that the equation X5 + X-1 = 0 has only one positive root?

Only one positive root means that only one root is greater than zero. If we assume that there are two or more positive roots, the conditions of Rolle's theorem must be satisfied in x > 0. Therefore, Rolle's theorem holds. If it doesn't hold, it means that our hypothesis doesn't hold, and the proposition is proved.

X minus 75 percent equals 0.5

x-75%=0.5
x=0.5-0.4;
x=0.1;
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3（X-5）^2=2（5-X）
4X ^ 2 + (radical 3 + radical 2) x + radical 6 = 0

3（X-5）^2=2（5-X）
3(x-5)^2-2(5-x)=0
3(x-5)^2+2(x-5)=0
(x-5)[3(x-5)+2]=0
(x-5)(3x-13)=0
X = 5 or x = 13 / 3
4X ^ 2 + (radical 3 + radical 2) x + radical 6 = 0
Is there a problem with this? Did you miss 2

How many hours is 48 minutes

48/60=4/5

25-4x + 3.2 = 8.4

25-4x+3.2=8.4
4x=25+3.2-8.4
4x=19.8
x=19.8÷4
x=4.95

Calculate the following questions. Those who can calculate simply should calculate 35 * 12 + 3500 / 70 (2.8 + 3.85 / 3.5) * 4.6
The simpler the process, the better

35*12+3500/70
= 35*12 + 50
= 35*(10+2) +50
= 35*10 +35*2 +50
= 350+70+50
= 470
（2.8+3.85/3.5）*4.6
= (2.8 + 1.1)*4.6
= 3.9 *4.6
= (4-0.1 )*3.6
= 4*3.6 -0.1*3.6
= 14.4 - 0.36
= 14.04

It is known that a, B and C are the three internal angles of the triangle ABC, and satisfy 2sinb = Sina + sinc. Let B be the maximum of Bo
It is known that a, B and C are the three internal angles of the triangle ABC, and satisfy 2sinb = Sina + sinc, let the maximum value of B be Bo. (1) find the size of Bo. (2) when B = (3BO) / 4, find the value of Cosa COSC

1、∵2sinB=sinA+sinC=2sin[(A+C)/2]cos[(A-C)/2]=2sin[(π-B)/2]cos[(A-C)/2]=2cos(B/2)cos[(A-C)/2]∴4sin(B/2)cos(B/2)=2cos(B/2)cos[(A-C)/2]∴sin(B/2)=(1/2)cos[(A-C)/2]≤1/2∴B/2≤π/6∴B≤π/3∴B0=π/32...

There is such a natural number. If you use it to remove 63, 91, 129, the sum of the three remainders is 28. What is the natural number?

63+91+129-28=255
255=3*5*17
It could be 15, 17 or 51
After examination, 17 are in accordance with the meaning of the question
So the natural number is 17