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Given the function f (x) = x + m / x, and the function image passes through the point (1,5), judge the monotonicity of function f (x) on [2, + ∞), and prove your conclusion with the definition
Let f (x) in (1,5) generation f (x), 1 + M = 5, M = 4, so f (x) = x + 4 / X definition, any x1, X2 belong to [2, + ∞], and X10 △ y = f (x2) - f (x1) = x2 + 4 / x2-x1-4 / X1 = x2-x1 + 4 (x1-x2) / x1x2 = (x2-x1) (x1x2-4) / x1x2-x1 > 0, x1x2 > 0, x1x2-4 > 0, so △ Y > 0, that is, the function f (x) is in [2, + ∞)
The function f (x) = x + m / X is known, and the function image passes through points (1,5). The monotonicity of function f (x) on [2, + ∞) is discussed and proved by definition. It is better to write down the draft paper and send pictures
F (x) = 2x / x ^ 2 + 1 of the parity, monotonicity, draw the function image, find the function maximum
F (- x) = - 2x / (x ^ 2 + 1) = - f (x) is an odd function F & # 39; = & nbsp; (2 (x ^ 2 + 1) - 2x (2x)) / (x ^ 2 + 1) ^ 2 & nbsp; = & nbsp; 2 (1-x ^ 2) / (x ^ 2 + 1) ^ 2 & nbsp; = 2 (1-x) (1 + x) / (x ^ 2 + 1) ^ 2, so [- 1,1] monotone increasing (- infinity, - 1) & nbsp; (1, + infinity) monotone decreasing maximum f (1) = 1 minimum f (- 1) = - 1
Let a be a matrix of order 3, A1 and A2 be the eigenvectors of a belonging to eigenvalue-1 and 1 respectively, and A3 satisfy the condition that aa3 = A2 + a3. It is proved that A1, A2 and A3 are linearly independent
It is proved that let k1a1 + k2a2 + k3a3 = 0 (1)
Then k1aa1 + k2aa2 + k3aa3 = 0
It is known that - k1a1 + k2a2 + K3 (A2 + a3) = 0
That is - k1a1 + (K2 + K3) A2 + k3a3 = 0 (2)
(1)-(2):2k1a1-k3a2 = 0
Because A1 and A2 are the eigenvectors of a which belong to the eigenvalues - 1 and 1 respectively,
So A1 and A2 are linearly independent
So K1 = K3 = 0
Substituting (1) to know K2 = 0
So A1, A2, A3 are linearly independent
What is the solution of X Λ 2-7x + 9 = 0?
a=1,b=-7,c=9,
△=b^2-4ac=49-36=13,
x=(7±√13)/2
Let two independent random variables X and y have the same distribution rate, and the distribution rate of X is x 0 1 P & nbsp; 12 & nbsp; 12, then the distribution rate of random variable z = max {x, y} is x 0 1 P & nbsp; 12______ ．
The combination of solutions (x, y) has the following four cases: (0,0), (0,1), (1,0), (1,1) the corresponding probabilities are 14. For the latter three cases, z = 1, for the first case, z = 0. Therefore, the distribution law of Z is Z = 0, P = 14, z = 1, P = 34
If x + MX-15 = (x + 3) (x + n), what is the value of M
x²+mx-15=x²+(n+3)x+3n
∴m=n+3
-15=3n
∴m=-2
n=-5
In the isosceles trapezoid ABCD, AD / / BC, angle B = 45 degrees, AE vertical, BC and point E, AE = ad = 2, then the trapezoid median line is
The triangle Abe is an isosceles right triangle trapezoid, the upper bottom is 2, and the lower bottom is composed of three parts. The corresponding part of AE's upper bottom and the lower bottom corresponding to AE's right side are 2 + 2 + 2 equal to 6, so the median line is 2 + 6 divided by 2 equal to 4
If the power of 2n + 1 + 3 of 3 is 324, try to find the value of n
3 ^ (2n + 1) + 3 ^ (2n) = 324 3 * 3 ^ (2n) + 3 ^ (2n) = 324 4 * 3 ^ (2n) = 324 3 ^ (2n) = 81 3 ^ (2n) = 3 ^ 4 2n = 4 N = 2 how does the power of 4 * 3 in the third step become 2n
Taking 3 ^ (2n) as a whole, a is 3A + a = 324, that is 4A = 324 * 3 ^ (2n) = 324
Sum Sn = 1 + 11 + 111 + +11…… 11 (n 1)
In this way, you want SN to be the first n terms of sequence an and a (n) = a (n-1) + 10 ^ (- n) a (2) = a (1) + 10 ^ (- 2) a (3) = a (2) + 10 ^ (- 3) a (4) = a (3) + 10 ^ (- 4)
RELATED INFORMATIONS
- 1. Judge and prove the monotonicity of function f (x) = X-1 / x + 2 on (2, positive infinity), and draw its general image
- 2. We know that the function f (x) = 2x ^ 2 + x-k, G (x) = ax ^ 3 + BX ^ 2 + CX + D (a is not equal to 0) is an odd function on R. when x = 1, G (x) obtains the extremum - 2 (1) finds the monotone interval and the maximum of the function; (2) if f (x) is less than or equal to (x) for any closed interval from - 1 to 3, finds the range of real number K 2) If f (x) is less than or equal to G (x) for any closed interval where x belongs to - 1 to 3, find the range of real number K
- 3. 1. The function y = ax ^ 2 + BX + C (a is not equal to 0) is a necessary and sufficient condition for an image to cross the origin_ 2. Known proposition p: "inverse ax belongs to R, inverse EM belongs to R, so that 4 ^ X-2 1. The function y = ax ^ 2 + BX + C (a is not equal to 0) is a necessary and sufficient condition for an image to cross the origin 2. Given proposition p: "inverted ax belongs to R, inverted EM belongs to R, so that 4 ^ X-2 ^ (x + 1) + M = 0", if proposition p is a false proposition, then the value range of real number m is
- 4. If the function f (x) = ax + 2 (a > 0 and a is not equal to 1) passes through a (2,6), then f (3) =?
- 5. It is known that the function f (x) = the cube of X + the square of AX + 3bx + C (B is not equal to 0), and G (x) = f (x) - 2 is an odd function?
- 6. If f (x + 1) = x & # 178; - 2x-15, then f (16) =?
- 7. ① Given f (2x) = 2x + 3, find f (x) 2. Given g (x) = 1-2x, f [g (x)] = 1-x & # 178; / X & # 178;, find the value of F (1 / 2)
- 8. Given that X and y satisfy X & # 178; + Y & # 178; + 5 / 4 = x-2y, what is the value of 2x + y?
- 9. Given X & # 178; + x = 1, find the value of X & # 179; + 2x & # 178; = 891 2. Proof: for any natural number n, the value of n (n + 5) - (n-3) (n + 2) can be divided by 6
- 10. (x²-4)²-10(x²-4)+25
- 11. The graph of function f (x) defined on R has two symmetry axes x = A and x = B (a is not equal to b). Find the period of F (x)
- 12. If the function f (x) defined on R is symmetric about x = A and about point (B, 0), and a is not equal to B, the period of function f (x) is obtained
- 13. Try to discuss the number of zeros of function H (x) = f (x + 1) - G (x) in the interval (- 2,0) Given function f (x) = LG (x + 1) 1. If G (x) is an even function and G (x) = g (x + 2), when 0 ≤ x ≤ 1, G (x) = f (x), find the analytic expression of y = g (x) (x belongs to [- 2,0]; 2. Under the condition of (1), the number of zeros of the function H (x) = f (x + 1) - G (x) in the interval (- 2,0) is discussed
- 14. The zero point of function y = (AX-1) (X-2) is discussed
- 15. If y = f (x) satisfies f (a) · f (b) < 0 on [a, b], then y = f (x) has zero in (a, b) This sentence is wrong, but I want to know what is wrong
- 16. Given that a, B and C belong to R, the function f (x) = ax ^ 3 + BX ^ 2 + CX satisfies f (1) = 0 Let the derivative of F (x) be f '(x), and f' (0) * f '(1) > 0 Finding the range of C / A
- 17. The function f (x) = ax ^ 3 + BX ^ 2 + CX + 3 defined on R satisfies the following conditions 1. F (x) is a decreasing function on (0,1) and an increasing function on (1, + OO) 2. F '(x) is even function 3. The tangent of F (x) at x = 0 is perpendicular to the line y = x + 2 Finding the analytic expression of function y = f (x) Let g (x) = 4lnx-m, if x belongs to [1, e], Let g (x)
- 18. F (x) = x ^ 3 + BX ^ 2 + CX + D (x belongs to R). It is known that f (x) = f (x) - f '(x) is an odd function and f (1) = t (T)
- 19. We know that the function is an odd function on f (x) = AX3 + CX + D. when x = 1, we take the extremum - 2 and find the monotone interval and maximum of F (x)
- 20. For the function f (x), if there exists x0 ∈ r such that f (x0) = x0 holds, then x0 is called the fixed point of F (x). It is known that f (x) = AX2 + (B + 1) x + (B-1) (a ≠ 0) (1) When a = 1, B = - 2, find the fixed point of F (x)