F (x) is equal to the square of negative x plus two ax and G (x) is equal to (2a minus one) times x plus one. In the interval [1,2], they are both decreasing functions, and the range of a can be obtained

F (x) is equal to the square of negative x plus two ax and G (x) is equal to (2a minus one) times x plus one. In the interval [1,2], they are both decreasing functions, and the range of a can be obtained

Let me tell you:
f(x)=-x^2+2ax,
According to B ^ 2-4ac ≥ 0, a ≥ 0
The axis of symmetry of F (x) is x = a
If f (x) is a decreasing function in the interval [1,2], then a ≤ 1,
However, G (x) = (2a-1) x + 1 is a decreasing function in the interval [1,2],
2a-1
0 to 1 / 2: process?
If ax square plus B x minus 1 less than 0 is negative, 1 less than x less than 2, then how much a is equal to and how much B is equal to
Title: ax & # 178; + BX-1 ＜ 0, the solution is: - 1 ＜ x ＜ 2, finding a, B
The solution, obviously, x = - 1 and 2 are the two roots of ax & # 178; + BX-1 = 0,
Substituting - 1 and 2 into the equation, we get the following results:
a(-1)²+b(-1)-1=0
2²a+2b-1=0
The solution is: a = 1 / 2, B = - 1 / 2
The solution of ax square plus B x minus 1 less than 0 is negative 1 less than x less than 2
The solution of ax & # 178; + BX-1 = 0 is X1 = - 1, X2 = 2
ax²+bx-1=a(x+1)(x-2)=ax²-ax-2a
B = - A, and - 1 = - 2A
The solution is a = 1 / 2, B = - 1 / 2
Let a, B, C and d be real numbers. What is the value of 2007a + 5B + 8C / 2007x + 5Y + 8Z?
Let a, B, C, d be real numbers, if the square of a + the square of B + the square of C = 25, the square of X + the square of Y + the square of Z = 36, ax + by + CZ = 30, then find 2007a + 5B + 8C / 2007x + 5Y + 8Z, what is the value?
Let a, B, C and d be real numbers
(a^2+b^2+c^2)(x^2+y^2+z^2)>=(ax+by+cz)^2
Take the equal sign when a / x = B / y = C / Z
(a^2+b^2+c^2)(x^2+y^2+z^2)>=(ax+by+cz)^2
So 25 * 36 > = 30 ^ 2
Obviously, the equal sign here
So a / x = B / y = C / z > 0
So a ^ 2 / x ^ 2 = B ^ 2 / y ^ 2 = C ^ 2 / Z ^ 2 = (a ^ 2 + B ^ 2 + C ^ 2) / (x ^ 2 + y ^ 2 + Z ^ 2) = 25 / 36
So a / x = B / y = C / z = 5 / 6
So (a + B + C) / (x + y + Z) = A / x = B / y = C / z = 5 / 6
Then (2007a + 5B + 8C) / (2007x + 5Y + 8Z) = 5 / 6
From Cauchy inequality
(a^2+b^2+c^2)(x^2+y^2+z^2)>=(ax+by+cz)^2
Take the equal sign when a / x = B / y = C / Z
(a^2+b^2+c^2)(x^2+y^2+z^2)>=(ax+by+cz)^2
So 25 * 36 > = 30 ^ 2
Obviously, the equal sign here
So a / x = B / y = C / z > 0
So a ^ 2 / x ^ 2 = B ^ 2 / y ^ 2 = C ^ 2... Expand
From Cauchy inequality
(a^2+b^2+c^2)(x^2+y^2+z^2)>=(ax+by+cz)^2
Take the equal sign when a / x = B / y = C / Z
(a^2+b^2+c^2)(x^2+y^2+z^2)>=(ax+by+cz)^2
So 25 * 36 > = 30 ^ 2
Obviously, the equal sign here
So a / x = B / y = C / z > 0
So a ^ 2 / x ^ 2 = B ^ 2 / y ^ 2 = C ^ 2 / Z ^ 2 = (a ^ 2 + B ^ 2 + C ^ 2) / (x ^ 2 + y ^ 2 + Z ^ 2) = 25 / 36
So a / x = B / y = C / z = 5 / 6
So (a + B + C) / (x + y + Z) = A / x = B / y = C / z = 5 / 6
Then (2007a + 5B + 8C) / (2007x + 5Y + 8Z) = 5 / 6
What are the conditions for determining the range of values by discriminant method?
First of all, don't copy it. For example, if the problem has a column, for example, y = (1-x ^ 2) / (1 + x ^ 2), if you use the discriminant method, it will be - 1
From y = (1-x ^ 2) / (1 + x ^ 2)
（y+1）x^2+y-1=0 *
When the coefficient of the highest order term is uncertain, we must discuss when the coefficient is 0
1) When y + 1 = 0, that is, y = - 1, then * is 0 times x ^ 2 + (- 1) - 1 = 0,
This formula is obviously not true, so y is not equal to - 1
2) When y + 1 is not equal to 0, * is a quadratic function and a discriminant can be used
The solution is - 1
Y = (1-x ^ 2) / (1 + x ^ 2) = - 1 + 2 / (1 + x ^ 2) the range should be (- 1, 1). 2 / (1 + x ^ 2) cannot be equal to 0
The solution set of B ^ 2-4ac0 is r
(A) Sufficient condition, but not necessary condition
(B) Necessary condition, but not sufficient condition
(C) Necessary and sufficient conditions
(D) Non sufficient and non necessary conditions
Ask for detailed explanation
If the solution set of ax ^ 2 + BX + C > 0 is r
The opening is upward, a > 0
And the solution set of B ^ 2-4ac0 is r
Then there must be B ^ 2-4ac0
So it's not a sufficient condition
Choose B
The solution set of ax ^ 2 + BX + C > 0 is r
It shows that X has no real root
That is to say, there is only one case in which the solution set of B ^ 2-4ac0 is r, that is, a > 0 and △ 0. Only if two conditions are satisfied at the same time, the solution set is R. obviously, in this problem, the former can not deduce the latter, so it is not sufficient, and the latter is tenable, so the former must be tenable, that is, the latter can deduce the former, so it is necessary. ... unfold
First of all, we need to know that the solution set of the quadratic inequality ax ^ 2 + BX + C > 0 is R. there is only one case, that is, a > 0, △ 0. Only if two conditions are satisfied at the same time, the solution set is R. obviously, in this problem, the former can not deduce the latter, so it is not sufficient, and the latter is tenable. The former must be established, that is, the latter can deduce the former, so it is necessary. To sum up, it is a necessary and insufficient condition. Put it away
Why can we use discriminant method to find the range of values
Please explain with mapping knowledge
The domain of any function should be a set of nonempty numbers, so the original function should be regarded as an equation about X, and there should be a real number solution, so the range of y can be obtained
If the solution set of the quadratic inequality ax & sup2; + BX + 2 > 0 is (- 1 / 2, - 1 / 3), then the value of a + B?
Description - 14
- -
It's 1 / 3 sobbing, not my fault
Are you still there, please_
-B/A=-1/2+1/3=-1/6,2/A=(-1/2)*1/3=-1/6,
A = - 12, B = - 2, a + B = - 14 can be obtained
Description X1 + x2 = - B / a = - 1 / 2-1 / 3 = - 5 / 6 x1x2 = 2 / a = 1 / 6
A=12 B=-10 A+B=2
How to find the range of discriminant?
Why does x belong to R
Because functions are mapped one by one
That is, given an independent variable, it must correspond to a dependent variable
That is, given an X, there must be a Y corresponding to it,
therefore
That is to say, for finding the range of value with discriminant
Draw an expression about y, there must be a solution!
So the discriminant must be ≥ 0
x1=[-b+sqrt(b^2-4ac) ]/2a
x2=[-b-sqrt(b^2-4ac) ]/2a
As long as dert = B ^ 2-4ac > 0, x1, X2 have solutions.
Do you mean this?
If ax ^ 2 + BX + C ≥ 0 (a
If ax & # 178; + BX + C ≥ 0 holds, then a = b = 0, C ＞ 0 is not consistent with a ＜ B, and ② a ＞ 0, △ = B & # 178; - 4ac ≤ 0, then 4ac ≥ B & # 178; it is easy to know that m = (a + 2B + 4C) / (B-A) = [a · (a + 2B + 4C)] [a · (B-A)] = (A & # 178; + 2Ab + 4ac) / (AB-A & # 178;); m ≥ (A & # 178; + 2Ab + B & # 1
Using discriminant method to find the range,
If there is a scope for X in the title,
\Function definition: a mapping defined on a nonempty number set is called a function. Therefore, given an X, there must be a Y corresponding to it. Therefore, by using the idea of equation, the original function variable is transformed into an equation about y. this equation must have a solution, equivalent to Δ≥ 0