If a is an acute angle and Sina + cosa = 2 / 3, find the value of the sixth power of sina + the sixth power of cosa It's sorted out,

If a is an acute angle and Sina + cosa = 2 / 3, find the value of the sixth power of sina + the sixth power of cosa It's sorted out,

sinA^6+cosA^6=(sinA^3+cosA^3)^2-2(sinAcosA)^3
=[(sinA+cosA)(1-sinAcosA)]^2-2(sinAcosA)^3
=[2/3*(23/18)]^2+2*5^3/18^3
=0.7685

Given that f (x) = x + 2 / X (1), it is proved that f (x) is an increasing function on [√ 2, + ∞]. Try to find the minimum value of the function g (x) = (x 2 + 6) / √ (x 2 + 4)

Let f (x) = x + 2 / X (1) prove that f (x) is an increasing function on [√ 2, + ∞] and (2) try to find the minimum value of the function g (x) = (x] + 6) / √ (x 2 + 4). It is proved that ∵ f (x) = x + 2 / x, and its definition domain is x ≠ 0. When x > 0, f (x) = x + 2 / x > = 2 √ 2 ｛ function f (x) on x = √ 2, take the minimum value 2 √ 2  f (x) in

Let f (x) = x 2 + 2 + 3 / x, (x ∈ [2, + ∞))) 1. Prove that f (x) is an increasing function. 2. Find the minimum value of F (x)

f(x)?
It is f (x) = (x 2 + 2x + 3) / x = x + 3 / x + 2
(1) Proof:
Take x1, X2 in [2, + ∞)
Let 2 ≤ x1

If the function f (x) = {- x 2 + X (x > 0) ax 2 + X (x ≤ 0), f (x) is an odd function when what is the value of A

The domain of F (x) is r
If f (x) is an odd function, f (- x) = - f (x)
When x > 0, - x0, - x

Given the function f (x) = x / (x? - 1), X belongs to (- 1,1). It is proved by definition that f (x) is an odd function on (- 1,1) 1) It is proved by definition that f (x) is an odd function on (- 1,1) 2) It is proved by definition that f (x) is a minus function on (- 1,1) 3) Solution of inequality f (m-1) + F (m) < 0 for M

1) Function domain (- 1,1) is symmetric about origin
f（-x）=-x/[(-x)^2-1]=-x/(x^2-1)=-f(x)
Therefore, f (x) is an odd function on (- 1,1)
2) For any x1, X2 ∈ (- 1,1), x1

It is proved that f (x) = x / 1 + X? Is an increasing function defined on (- 1,1) It is proved that if x 1, x 2 are taken at any position (- 1,1), then X1 is assumed

x1(1+x2²）-x2(1+x1²)
=x1+x1x2²-x2-x2x1²
=(x1-x2)- x1x2(x1-x2)
=(1-x1x2)(x1-x2)

Given the function f (x) = x + 1 / X (1), it is proved that the function f (x) is an odd function (2) by definition, it is proved that the function f (x) is an increasing function on (1, + ∞) Given the function f (x) = x + 1 / X (1), it is proved that the function f (x) is an odd function (2) by definition, it is proved that the function f (x) is an increasing function on (1, + ∞)

(1) f(-x) = (-x) + 1/(-x) = -(x+ 1/x) = -f(x)
(2) This is a tick function
Let 1 f (x1) - f (x2) = (x1-x2) + 1 / x1-1 / x2 = (x1-x2) - (x1-x2) / (x1x2) = (x1-x2) [1-1 / (x1x2)]
Because of 1, it is proved

It is proved that (1) the function y = x 2 + 3x + 1 has two different zeros; (2) the function f (x) = x ^ 3 + X-1 has zero points on the interval (0,1)

1) Delta = 3 ^ 2-4 = 5 > 0, so y has two different zeros
2） f(0)=-10
So there must be zero on (0,1)

What is the value of F (2) if the function f (x+1) =3x 2 +5 is known

f(x+1)=3x²+5
Put x = 1 into the above formula to get
f(2)=3*1^2+5=8

This paper points out that the function f (x) is equal to the vertex coordinates of each image of 3x? And G (x) = 3x? - 3x + 1, and explains the similarities and differences between them 2, the function y = ax 2 + BX + C (a > 0, b)

The vertex coordinates of the 1 function f (x) = 3x 2 are o (0,0)
The vertex coordinates of G (x) = 3x? - 3x + 1 = 3 (x-1 / 2) 2 + 1 / 4 are m (1 / 2,1 / 4)
The common points of their images are: 3 > 0, the opening of the image is upward, and the image has the lowest point
Difference: the symmetry axis is different, (x = 0, x = 1 / 2), and the intersection point of X, Y axis is different
2. The vertex coordinates m (- B / (2a), (4ac-b 2) / (4a)) of the function y = ax 2 + BX + C (a > 0, B < 0, C < 0),
∵a>0,b<0,c<0
∴-b/(2a)>0,4ac-b²)/(4a)<0
ν m (- B / (2a), (4ac-b 2) / (4a)), in the fourth quadrant
3. (reverse thinking) y = f (x) = x? - 2x + 1 downward translation 3 units, y = H (x) = f (x) - 3 = x? - 2x-2
Y = H (x) = f (x) - 3 = x? - 2x-2 shifts two units to the right, and y = m (x) = H (X-2) = (X-2) 2 (X-2) - 2 = x? - 6x + 2
∴x²-6x+2=x²+bx+c
∴b=-6,c=2