How many real number solutions to the equation SiNx = x / 10 about x? The process is important

How many real number solutions to the equation SiNx = x / 10 about x? The process is important

There are seven solutions, as shown in Fig
By means of odd function and symmetry and combining with x = 10 in y = x / 10 when y = 1, the length range of image investigation is limited

Try to judge the number of real number solutions of the equation SiNx = X100 π

The number of real number solutions of the analytic equation SiNx = X100 π is equal to the number of image intersections of the functions y = SiNx and y = X100 π ∵ | SiNx | ≤ 1 | | X100 π | ≤ 1, | x | ≤ 100 π when x ≥ 0, as shown in the right figure, there are 100 intersections of the two lines, because y = SiNx and y = X100 π are odd functions

Excuse me: if the equation cos2x + cosx-a = 0 has a solution, then the range of real number a is

2(cosx)^2+cosx-1-a=0
(1) If the equation has a solution, the discriminant 1 ^ 2-4 * 2 * (- 1-A) ≥ 0
The solution is a ≥ - 9 / 8
(2) Because - 1 ≤ cosx ≤ 1
Then the product of two ∈ [- 1,1]
That is - 1 ≤ (- 1-A) / 2 ≤ 1
The solution is - 3 ≤ a ≤ 1
To sum up: - 9 / 8 ≤ a ≤ 1

If the equation 4x + a · 2x + 4 = 0 about X has a real solution, then the value range of real number a is______ ．

A = - 4x-42x, let 2x = t (T > 0), then - 4x-42x = - T2 + 4T = - (T + 4T) because t + 4T ≥ 4, so - 4x-42x ≤ - 4, so the range of a is (- ∞, - 4], so the answer is: (- ∞, - 4]

Linear Algebra: proving invertible matrices?
It is known that a, B and a + B are all invertible. It is proved that A-1 + B-1 is also invertible

A^－1+B^－1=A^-1(B+A)B^-1
So (a ^ - 1 + B ^ - 1) * [b (a + b) ^ - 1A] = E
And a, B, a + B are reversible,
So a ^ - 1 + B ^ - 1 is also invertible, and the inverse matrix is B (a + b) ^ - 1A

How to prove the property of mixed product in high number vector, that is, how to prove (a × b) · C = (a × C) · B = (B × C) · a

The final expression of mixed product (a × b) · C is a determinant: (a × b) · C = A1 A2 a3B1 B2 b3c1 C2 C3, so, (B × C) · a = B1 B2 b3c1 C2 c3a1 A2 A3 (C × a) · B = C1 C2 c3a1 A2 a3B1 B2 B3 according to the nature of determinant, the value of determinant changes sign

Three teachers led some students on a spring outing to negotiate with two travel agencies. The preferential terms offered by Shanshui company are: teachers pay in full, students pay 70% off; the preferential terms offered by Shenyou company are: all teachers and students charge 20% off. Which company is more economical?

Suppose that each student needs to pay X Yuan and there are y students. According to the meaning of the question, the fee of Shanshui company is: 3x + 70% XY. The fee of Shenyou company is: 80% (3 + y) X80% (3 + y) x - (3x + 70% XY) = 2.4x + 0.8xy-3x-0.7xy = (0.1y-0.6) X. answer: when there are 6 students, the two companies need the same fee. When the number of students is more than 6, Shanshui company saves money. When the number of students is less than 6, Shanshui company saves money , then Shenyou company saves money

Find Lim x tends to 0 (e ^ x-x-1)

The answer is 0, expand e ^ x with Taylor formula, e ^ x = 1 + X + O (x ^ n), O (x ^ n) is the infinitesimal of X, so when x tends to 0, the limit is 0

Given that the domain of F (2 ^ x) is [- 1.1], then the domain of F (log2x) is
Such as the title

Let 2 ^ x = m, m belong to [1 / 2,2], have
X=log2(m),-1≤x≤1.
-1 ≤ log2 (m) ≤ 1,1 / 2 ≤ m ≤ 2
1/2≤log2(x)≤2,
√2≤x≤4.
Then the definition field of F (log2x) is √ 2 ≤ x ≤ 4

Who do you thank

Who do you thanks?