The function f (x) = sin (2x + π / 3) is used to find the axis of symmetry, the center of symmetry and monotone interval

The function f (x) = sin (2x + π / 3) is used to find the axis of symmetry, the center of symmetry and monotone interval

sin(2x+π/3)=±1
2x+π/3=kπ+π/2
So the axis of symmetry is x = k π / 2 + π / 12
sin(2x+π/3)=0
2x+π/3=kπ
x=kπ/2-π/6
So the center of symmetry is (K π / 2 - π / 6,0)
When sin increases, 2K π - π / 2 < 2x + π / 3 < 2K π + π / 2
K π - 5 π / 12, so the increasing interval is (K π - 5 π / 12, K π + π / 12)
Similarly, the minus interval is (K π + π / 12, K π + 7 π / 12)

The monotone increasing interval of the function f (x) = SiNx + cosx is

The square of SiNx + cosx = 1 + 2sinxcosx = 1 + sin2x
When SiNx + cosx = 0
When x belongs to [3 / 4 π + 2n π, 7 / 4 π + 2n π]
F (x) = radical (1 + sin2x) increases by [- 1 / 4 π + n π, 1 / 4 π + n π]
So f (x) increases in the interval [3 / 4 π + 2n π, 5 / 4 π + 2n π]
To sum up, f (x) increases monotonically on [1 / 4 π + 2n π, 5 / 4 π + 2n π], n is an integer
Of course, it's OK to use derivative function, but it's about calculus

The function y = SiNx | SiNx | cosx | cosx | is the monotone decreasing interval?

PI 2 is a period of this function
When 0

Monotone decreasing interval of function y = (SiNx) ^ 4 + (cosx) ^ 4

(SiNx) ^ 4 + (cosx) ^ 4 = [(SiNx) ^ 2 + (cosx) ^ 2] ^ 2-2 (sinxcos x) ^ 2 = 1-1 / 2 (sin2x) ^ 2 = 1-1 / 2 (sin2x) ^ 2 = 1-1 / 2 * 1 / 2 * (1-cos4x) = 1-1 / 4 + 1 / 4cos4x = 1 / 4cos4x = 1 / 4cos4x + 3 / 42K π ≤ 4x ≤ (2k + 1) π, K ∈ Z: X ∈ [K π / 2, K π / 2 + π / 4], k k k k, K ∈ Z: X ∈ [K π / 2, K π / 4], K, K, K, K, K, K, K, K monotone reduction when ∈ Z

The function y = | SiNx | + | cosx | is the monotone decreasing interval?

{x|Kπ-(π/2)

Find the maximum value of the function y = cos ^ 2x SiNx, and find the corresponding x value when the function gets the maximum value

From the meaning of the title: y = 1-2 (SiNx) ^ 2-sinx = - 2 (SiNx + 1 / 4) ^ 2 + 9 / 8
Because: - 1 〈 = SiNx

Find the maximum value of the function f (x) = (SiNx + COS) ^ 2 + 2cos ^ x on X ∈ [- π / 2, π / 4]

F (x) = (SiNx + COS) ^ 2 + 2cos ^ 2x = 1 + 2sinxcosx + cos2x + 1 = 2 + sin2x + cos2x = 2 + √ 2Sin (2x + π / 4) ∵ x ∈ [- π / 2, π / 4] ∵ 2x + π / 4 ∈ [- 3 π / 4,3 π / 4], then sin (2x + π / 4) ∈ [- 1,1]

Let f (x) = root 2 of log 2 (x of root 2 sin 2), (1) define the intersection of the minimum value of domain (2) and X axis (3)

1. The domain is SiNx / 2 > 0
2K π, so 4K π 2. minimum value: because the base number = root sign 2/2<1
So it's a minus function. When the real number is the largest, it takes the minimum value
The maximum value of root 2 (SiNx / 2) is root 2
So the minimum value is f (x) = log (root 2 / 2) ^ root 2
3. Intersect with X axis, i.e. y = 0, so the real number = root 2 (SiNx / 2) = 1
So SiNx / 2 = radical 2 / 2
x/2=2kπ+π/4
x=4kπ+π/2

If the function f (x) satisfies f (2 / (x - | x |) = log2, then the analytic expression of F (x) is

Since the absolute values of - X and X are under the root sign, - x > 0, so x

Given the root 2 ≤ x ≤ 8, find the maximum and minimum values of the function f (x) = (log2x / 2) (log24 / x)

f(x)=(log2x-log22)(log24-log2x)
=(log2x-1)(2-log2x)
Let t = log2x
F (x) = (t-1) (2-T)
=-t^2+3t-2
=-(t-3/2)^2+1/4
So ymax = 1 / 4
ymin=5/2