What are the correct propositions related to the function f(x)=sin(x)x?

Given the function fx sinxx, which of the following statements is correct

It is known that the function f(x)=sinx/x, which of the following propositions is correct

1. f(x) is an odd function

②For any x in the definition domain, f(x)

③When x=3π/2, f(x) obtains the minimum value;

④f(2)>f(3)

5. When x>0, if the absolute value of the equation f(x)=k has and has only two different real solutions α, β (α>β), then β*cosα=-sinβ

Analysis: ∵ function f(x)=sinx/x, its domain is x≠0

f(-x)=-sinx/(-x)=f(x)==>even function;

∴（1）Wrong

∵When x tends to 0, the limit of function f(x) is 1

∴In the domain of definition f(x)

∴（2）Correct

When x>0, f'(x)=(xcosx-sinx)/x^2

f'(3π/2)=(0 1)/(3π/2)^2≠0

∴（3）Wrong

∵When x tends to 0, the limit of function f(x) is 1, f(π)=0

∴The function decreases monotonically on the interval (0, π]; ==>f(2)>f(3)

∴（4）Correct

When x>0,

When X∈(0, π), f(x)>0,

When X∈(π,2π), f(x)

takes the absolute value and becomes k

∵The absolute value of equation f(x)=k has and only two different real solutions α, β (α>β)

∴cosα=-k

f（β）=sinβ/β

∵k=f（β）=sinβ/β==>-cosα*（β）=sinβ

∴（5）Correct

To summarize: 2, 4, and 5 are correct

Known function fx Asinωx φ A 0 ω

(Ⅰ) From the image, we know that A=2, the minimum positive period of f(x) is T=4*(

5π

12 -

π

6 )=π，∴ω=2

point (

π

6,2) Substitute to get sin(

π

3 φ)=1, and |φ|π

2, ∴φ=

π

6

So the analytical formula of function f(x) is f(x)=2sin(2x

π

6 )

（Ⅱ）g(x)=2sin(2x

π

6 )-2cos2x=

3 sin2x-cos2x=2sin(2x-

π

6 )

The transformation is as follows: translate the image of y=sinx to the right

π

6 Get y=sin(x-

π

6 ); then sin(x-

π

6 )

The abscissa coordinates of all points on the image are shortened to the original

1

2 If the vertical coordinate remains unchanged, we get y=sin(2x-

π

6) image;

Put y=sin(2x-

π

6) The vertical coordinates of all points on the image are expanded to twice the original size, and the horizontal coordinates remain unchanged to obtain y=2sin(2x-

π

6 ) image.

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