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Given that the function f (x) is even function, f (x) is differentiable, it is proved that f '(x) is an odd function
According to the definition of derivative: F '(- x) = LIM (a - > 0) [(f (- x + a) - f (- x)) / a] = LIM (a - > 0) [f (x-a) - f (x)] / a = - F' (x)
It is proved that if f (x) is a derivative odd function on R, then f '(x) is even on R
F '(- x) = Lim [f (- x + ⊿ x) - f (x)] / ⊿ x, ⊿ x, ⊿ x, ⊿ x approaches to zero limit f' (- x) = Lim [f (- x + ⊿ x) - f (x)] / X, change ⊿ x to - ⊿ x, we get f '(- x) = [f (- X - ⊿ x) - f (x)] / - 88; x = [- f (x + ⊿ x) + F (x)] / - X x = [f (x + ⊿ x) - f (x)] / ⊿ x = f' (x), so it is even even even even, so it is even even even even, so it is even even even even even function
Among the following functions, a.y = 2 ^| x | b.y = 2 ^ x + 2 ^ (- x) C.y=lg(1/x+1) DLG [x + radical (x ^ 2 + 1)] Also, I think D is also a non odd and non even function, because its definition domain is not symmetric about the origin ~ teach me!
x+√（x^2+1) >0
prove
Because √ (x ^ 2 + 1) > √ x ^ 2
Therefore, when x > 0, x + √ (x ^ 2 + 1) > 0
When x √ x ^ 2 √ (x ^ 2 + 1) > - x x x + √ (x ^ 2 + 1) > 0
So the definition field of the D option is that all real numbers are symmetric about the origin, and √ is the root sign
f(-x)=lg[-x+√（x^2+1)]=lg 1/[x+√（x^2+1)] =lg1-lg[x+√（x^2+1)] =0-f(x)=-f(x)
So D is an odd function
This is the use of [x + √ (x ^ 2 + 1)] * [- x + √ (x ^ 2 + 1)] = [√ (x ^ 2 + 1)] ^ 2-x ^ 2
=x^2+1-x^2=1
Among the following functions, a y = x + 1 b y = - x ^ 2 c y = 1 / x d y = x|x|
Even if a function is odd or even, it must satisfy f (- x) = f (x) = - f (x), so it can only have f (x) = 0
So a, B, C, D are not
Where a is non odd and non even; B is even; C is odd; D is odd
An isosceles triangle is divided into 4 equal area triangles with 3 lines. How to divide them?
Divide the base of the isosceles triangle into four parts, so that there are three bisection points, respectively connecting the three bisection points and the vertex. In this way, the isosceles triangle is divided into four triangles with the same figure area by three lines
Take a point P in the square, divide it into four isosceles triangles (except diagonal), and find three kinds of division emergency
There are five methods, as shown in the figure
How to draw an isosceles triangle into four equal area triangles?
The easiest way is to make four 4 equal points on the bottom and connect them. Because the height is the same, the side length is also. So the area is equal
How to divide an isosceles triangle into 4 equal parts Speed is needed
Divide the bottom into four equal parts
How can a square be divided into four equal isosceles triangles It is as like as two peas of four isosceles.
Draw a diagonal line, exactly 4 isosceles triangle
There are several ways to divide an isosceles triangle into three isosceles triangles The contents are as follows, I asked a lot of people about this question, and they all said it was one, I don't think so
4 kinds
Make two bisectors of the angle between the waist and the base
Take the midpoint of each side and connect again
There is one that can be determined by the degree of the angle
The intersection of the perpendicular lines
RELATED INFORMATIONS
- 1. How to divide a triangle into four parts It's not equilateral
- 2. One angle of an isosceles triangle is 50 degrees. Find the degrees of the other two angles In an isosceles triangle, the top angle is four times of the base angle
- 3. Given that one of the interior angles of an isosceles triangle is 80 degrees, then the degrees of the other two inner angles are () A. 50°，50° B. 80°，20° C. 50, 50 or 80, 20 D. None of the above is true
- 4. Given that an interior angle of an isosceles triangle is 80 degrees, then the top angle of the isosceles triangle is () degrees
- 5. If the top angle of an isosceles triangle is x degrees, then the angle between the high line on the waist and the bottom edge is______ Degree
- 6. The ratio of a base angle to a vertex degree of an isosceles triangle is 1:4. What is the top angle of this isosceles triangle?
- 7. If the height of the base edge of an isosceles triangle is equal to half of the waist length, then the base angle of the isosceles triangle is equal to______ Degree
- 8. It is proved that the height on the base of an isosceles triangle with a vertex angle of 120 degrees is equal to half of the waist length May have to discuss the classification, a good solution to give more points
- 9. As shown in the figure, it is known that in isosceles △ ABC, the vertex angle ∠ a = 36 ° and BD is the bisector of ∠ ABC The value of AC is equal to () A. 1 Two B. 5−1 Two C. 1 D. 5+1 Two
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- 14. As shown in the figure, it is known that y = f (x) is an even function defined on R. when x ≥ 0, f (x) = x2-2x (1) Find the values of F (1) and f (- 2); (2) Find the analytic formula of F (x) and draw a diagram; (3) The root of equation f (x) = k is discussed
- 15. It is known that f (x) is an even function defined on R, and f (x + 1) = f (x-1) holds for any real number x, when x belongs to [1,2], f (x) =?
- 16. If f (x) is an even function and f '(0) exists, it is proved that f' (0) = 0
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- 18. Let f (x) be defined on (- L, l), prove that f (x) + F (- x) is even function, f (x) - f (- x) is odd function
- 19. It is known that the function f (x) whose domain of definition is R is a decreasing function on (8, + ∞) and the function y = f (x + 8) is even, then () A. f（6）＞f（7） B. f（6）＞f（9） C. f（7）＞f（9） D. f（7）＞f（10）
- 20. Given the function f (x) = 2Sin (2x + π / 6), find 1. Find the minimum positive period of the function; 2. Find the value range of the function f (x) when x ∈ [0, π / 2]; (3) find the monotone decreasing interval of F (x) when x ∈ [- π. π]