The monotone increasing interval of function f (x) = (radical 2-1) ^ (x ^ 2-3x + 2) is

The monotone increasing interval of function f (x) = (radical 2-1) ^ (x ^ 2-3x + 2) is

The original function is composed of y = (radical 2-1) ^ t and T = x ^ 2-3x + 2
Y = (radical 2-1) ^ t is monotonically decreasing,
The function T = x ^ 2-3x + 2 = (x-3 / 2) ^ 2-1 / 4 decreases monotonically on (- ∞, 3 / 2],
According to the principle of the same increase but different decrease of the composite function,
(2) = (2) = (2) = ∞

The symmetric axis of the function f (x) = 2Sin (Wx + π / 4) (W > 0) is exactly the same as that of the function g (x) = cos (2x + φ) (│φ≤ π / 2) Then what is the value of φ,

The period is the same, w = 2
2X + π / 4 = (2x + φ) + π / 2 + K π
And because │φ≤ π / 2
So φ = - π / 4

It is known that the symmetric axes of the images of the functions f (x) = 2Sin (ω X - π / 6) (ω > 0) and G (x) = cos (2x + φ) - 3 are identical. If x belongs to [π / 3, π / 6], Then the value range of F (x) is?

It is known that the symmetric axes of the images of the functions f (x) = 2Sin (ω X - π / 6) (ω > 0) and G (x) = cos (2x + φ) - 3 are identical. If x belongs to [- π / 3, π / 6], then what is the range of F (x)?
Let ω X - π / 6 = π / 2 + K π, the symmetry axis of F (x) is x = 2 π / 3 ω + K π / ω;
Let 2x + φ = k π, that is, the symmetry axis of G (x) is: x = - φ / 2 + K π / 2;
The symmetry axes of the two are exactly the same, that is, 2 π / 3 ω + K π / ω = - φ / 2 + K π / 2;
SO 2 π / 3 ω = - φ / 2. (1); K π / ω = k π / 2. (2)
From (2) we get ω = 2; by substituting formula (1), we get φ = - 2 π / 3
When - π / 3 ≤ x ≤ π / 6, minf (x) = f (- π / 4) = 2Sin (- π / 2 - π / 6) = - 2cos (π / 6) = - √ 3
maxf(x)=f(π/6)=2sin(π/3-π/6)=2sin(π/6)=1.

It is known that the symmetric axis of the image of F (x) = 2Sin (Wx + π / 6) is exactly the same as that of G (x) = cos (3x + FAI) + 2 if x ∈ (0, π / 9) What are the minimum and maximum values of F (x)?

The same axis of symmetry means the same period, so w = 3
F (x) = 2Sin (3x + π / 6)
0＜x＜π/9
π/6＜3x+π/6＜π/2
So 1 / 2 < sin (3x + π / 6) < 1
The minimum value of F (x) is 1 and the maximum value is 2

It is known that the symmetry axes of the image with F (x) = 2Sin (Wx + π / 6) and G (x) = cos (3x + a) are exactly the same. How to judge whether the value of W =? A can be obtained? A =? But the answer book "according to the symmetry axis is exactly the same, w = 3" is not very understood

2Sin (Wx + π / 6) = KCOs (3x + a) (k is not equal to 0)
Cos function is shifted from sin function to left (or right) by 90 degrees (i.e. π / 2)
And the symmetry axis of F (x) = 2Sin (Wx + π / 6) and G (x) = cos (3x + a) are exactly the same. I think the period is the same, and the period is related to X and the coefficient before it
So a = π / 6 + π / 2 should be independent of W

The definition domain, range, period, symmetry center, symmetry axis, monotone interval and the set of X at the maximum value of y = 2Sin (2x + π / 4) For example, we also have the definition domain of y = 2cos (2x + π / 4), the range of values, the period, the center of symmetry, the axis of symmetry, the monotone interval, and the set of X at the maximum value

Y = 2Sin (2x + π / 4) y = 2cos (2x + π / 4) definition domain r r r range [- 2,2] [- 2,2] period π π symmetry center (K π / 2 - π / 8,0) (K π / 2 + π / 8,0) symmetry axis X = k π / 2 + π / 8 x = k π / 2 - π / 8 monotone interval increase (K π - 3 π / 8, K π + π / 8) (K π +...)

Given the function y = 2Sin (2x + π / 4) + 1, find 1. Period 2. The maximum value and the set of X angle when obtaining the maximum value. 3. Monotone interval. 4. Symmetry axis and symmetry center

Don't be afraid to see the formula in sin
Sin (2x + π / 4) is in the final analysis the form of sin T, but t = 2x + π / 4
Don't talk nonsense and solve problems
The periodic formula t = 2 π / W
W is the coefficient in front of X
So t = π
The maximum value is obtained when sin (2x + π / 4) = 1. The maximum value is 3
Solve the equation. 2x + π / 4 = sk π + π / 2
X = k π + π / 8. K integers
The minimum value is - 1 when sin (2x + π / 4) = - 1
Solve this equation. 2x + π / 4 = sk π - π / 2
x=kπ-3π/8
Monotone interval:
SiNx increases monotonically at x = [2K π - π / 2,2k π + π / 2]
So sin (2x + π / 4): 2x + π / 4 = [2K π - π / 2,2k π + π / 2] increases monotonically
x=[kπ-3π/8,kπ+π/8]
SiNx decreases monotonically at x = [2K π + π / 2,2k π + 3 π / 2]
So sin (2x + π / 4): 2x + π / 4 = [2K π + π / 2,2k π + 3 π / 2] decreases monotonically
x=[kπ+π/8,kπ+5π/8]
Axis of symmetry:
The symmetry axis of SiNx is the wave crest and trough. X = k π + π / 2
So the axis of symmetry of sin (2x + π / 4) is 2x + π / 4 = k π + π / 2
x=kπ/2+π/8
The opposite center of SiNx is the point of SiNx = 0. X = k π
So the symmetry center of sin (2x + π / 4) is 2x + π / 4 = k π
x=kπ/2-π/8
Conclusion: please take a close look at my steps. No matter how complex the algebraic expression in sin () is, standing outside sin () will still be
A sine function. Just think of the algebraic expression as the X of the standard sine function

Find the definition domain, range, period, symmetry axis, symmetry center, maximum value of function y = 2Sin (2x - π / 3) and obtain the corresponding x value of the maximum value Online, etc., correct second batch

The definition domain of y = 2Sin (2x Pai / 3) is r, the range of value is [- 2,2], the minimum positive period T = 2pai / 2 = Pai, the symmetry axis is 2x Pai / 3 = KPAI + Pai / 2, that is, x = KPAI / 2 + 5pai / 12, the symmetry center is 2x Pai / 3 = KPAI, that is, x = KPAI / 2 + Pai / 6, i.e., the center of symmetry is (KPAI / 2 + Pai / 6,0), and the maximum value is 2

Function y = 2Sin (3x + π / 4) 1 definition domain 2 range 3 symmetry center 4 symmetry axis 5 monotone increasing interval 6 monotone decreasing interval

1, define the domain as R
2, the range is [- 2,2]
3, symmetry center (2k π / 3 - π / 12,0)
4, axis of symmetry x = k π / 3 + π / 12
5, monotonically increasing interval (2k π / 3 - π / 4,2k π / 3 + π / 12)
6, monotone decreasing interval (2k π / 3 + π / 12,2k π / 3 + 5 π / 12)

Let f (x) = sin (2x + Φ) (- π)

Because the function is symmetric about this line,
According to the characteristics of this function, it can be determined that the function f (x) = 1 or - 1 when x = π / 8,
So 2x + φ = k π + π / 2 (k is a real integer),
That is, π / 4 + φ = k π + π / 2,
So φ = k π + π / 4,
And - π < φ < 0, so K can only be taken as - 1,
So φ = - 3 π / 4
The original equation is f (x) = sin (2x-3 π / 4), and the monotone increasing range of SiNx is [2K π - π / 2,2k π + π / 2],
So 2K π - π / 2 < 2x-3 π / 4 < 2K π + π / 2,
The monotone interval of F (x) is [K π - π / 8, K π + 5 π / 8], and K is a real integer