Ask a question about periodic function F (x) is a bounded function on R, f (x + 13 / 42) + F (x) = f (x + 1 / 6) + F (x + 1 / 7), find the smaller positive period of F (x) The answer in the reference book is 1 / 42, but the process is wrong. I can only prove that 1 is the period of F (x) How to find out that 1 / 42 is f (x) period,

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• 2. A problem about periodic function (see supplement) Given the function f (x) = sinwx coswx, if there is a real number X1 such that f (x1) ≤ f (x) ≤ f (x1 + 4) holds for any real number x, what is the minimum value of positive number W? The answer is π / 4. I mainly want to know why it is enough to take half a period between the maximum and the minimum. I always feel that for the interval (K π, K π + π / 2), there is exactly half a period, but not any f (x) has a function value corresponding to this interval Why is half a period between F (x1) and f (x1 + 4) enough? Does it have to be the maximum and minimum?
• 3. On the mathematical problems of periodic function Given that the even function f (x) defined on R satisfies f (x) = - f (4-x) and when x ∈ [2.4], f (x) = log2 (x-1) [2 is the base, X-1 is the real number], what is the value of F (2010) + F (2011)? By the way, since he said f (x) is an even function, is there f (4-x) = f (x-4) and why?
• 4. A mathematical problem, periodic function If f (x) is a function defined on the set of real numbers and f (x + 2) = f (x + 1) = f (x), f (1) = Lg3 / 2, f (x) = LG15 Prove by definition that it is a periodic function
• 5. The problem of periodic function in Mathematics If the function y = f (x) satisfies for any real number x in the domain of definition, 1. F (x + a) = - f (x), then y = f (x) with T=_____ It is a period 2. F (x + a) = 1 / F (x), then y = f (x) with T=_____ It is a period 3. F (x + a) = - 1 / F (x), then y = f (x) with T=_____ It is a period 4. F (x + a) = [1 + F (x)] / [1-f (x)], then y = f (x) with T=_____ It is a period 2a 2.2a 3.2a 4.2a But how did it come about?
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• 8. Can we give the operational properties of function period in detail For example, f (x) = f (x + T) or other In addition, is f (x-a) = f (x + a) a periodic operation?
• 9. How to prove the periodic problem of two functions? If f (x) is an odd function and the equation f (a + x) = f (A-X) holds for all x ∈ R, it is proved that the period of F (x) is 4a If f (x) is symmetric with respect to (a, Y0) and x = B, it is proved that the period of F (x) is 4 (B-A)
• 10. Two questions about the proof of periodic function 1. It is known that f (x) is an odd function, and the image of F (x) is symmetric with respect to the line x = 2. It is proved that f (x) is a periodic function 2. Let f (x) be an even function defined on R whose image is symmetric with respect to the line x = 1. For any x 1, x 2 belonging to [0,0.5], f (x 1 + x 2) = f (x 1) * f (x 2) is proved to be a periodic function
• 11. A problem about the period of function If f (x) is an odd function on R and an increasing function on (- 1,0), and f (x + 2) = - f (x), find the period of F (x)? The answer is f (x) = - f (x + 2) = f (x + 2 + 2) = f (x + 4). Why can't f (x) = - f (x + 2) be used directly, and the period is 2
• 12. A problem of periodic function, Given that the domain of definition of function y = f (x) is r, f (x + 2011) = f (x + 2010) + F (x + 2013) holds for any real number X. if f (1) + F (2) = 1 and f (1) + F (2) + +If f (2013) = 0, then f (2007) =?
• 13. It is known that the parabola y = MX2 - (3m + 4 / 3) x + 4 intersects the X axis at points a and B, and intersects the Y axis at point C (1) Find the coordinates of three points a, B and C (one of them can be expressed by an algebraic expression containing m) (2) If △ ABC is an isosceles triangle, find the analytical formula of parabola There are four situations in the second question
• 14. A proof of mathematical function Let an implicit function satisfy f (x, y) = f (y, x) = K (k is a constant) It is proved that the explicit function y = f (x) determined by F (x, y) = k must satisfy f (x) = F-1 (x)
• 15. A mathematical function problem If the line y = (1 + M & sup2;) x + B intersects the X axis at P (- 5,0), then___ When y ＞ 0___ Y < 0
• 16. Several mathematical function problems 1. Function y = the square of X / the square of X + 1, X belongs to real number, find the function range 2. The maximum value of function y = 1 / 1-x (1-x) 3. If the square of function y = x + AX-1 has the minimum value - 2 in the interval [0,3], then the real number a is equal to? 4.2lg (X-Y) = lgx + lgY, then Y / x equals?
• 17. The known function f (x) = 3-4asinxcosx + 4cos & sup2; x-4 (cosx) ^ 4 1. If a = 1, find the maximum and minimum of F (x) 2. If the minimum value of function f (x) is 1, find the value of A
• 18. Test questions of mathematical function in grade one of senior high school Given that there is a point P (- radical 3, m) on the terminal edge of angle θ, and sin θ = 4 / 4 radical 2 m, the value of cos θ = Tan θ is obtained
• 19. If there is no real element in the solution set of equation x ^ 2-2x + 2A = 0, find the straight range of real element a
• 20. Is that a period for a mathematical function? Which is the image?