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Given the function f (x) = (1 / 3) ^ x, X belongs to negative 1 to 1, and the minimum value of function g (x) = (f (x)) ^ 2-2af (x) + 3 is h (a), the analytic expression of H (a) is obtained Solution: f(x)=(1/3)^x ,x∈(-1,1) ∴f(x)∈(1/3,3) Let t = f (x) ∈ (1 / 3,3) ∴g(t)=t²-2at+3 I can't understand it from here on The symmetry axis of G (T) is t = a (1) When a > 3, G (T) min = g (3) = 9-6a + 3 = 12-6a (2) When a < 1 / 3, G (T) min = g (1 / 3) = 1 / 9 - (2 / 3) a + 3 = 28 / 9 - (2 / 3) a (3) When a ∈ [1 / 3,3], G (T) min = g (a) = A & # 178; - 2A & # 178; + 3 = 3-A & # 178; G (T) symmetry axis is t = A. why do we get the following three classification discussions
Because g (T) is a quadratic function, it is transformed into G (T) = (T-A) &# 178; + 3-A & # 178;, and the axis of symmetry is a. because t ∈ (1 / 3,3), when a > 3, G (T) monotonically decreases in 1 / 3,3), so the minimum value of G (T) is g (3). Similarly, when a < 1 / 3, G (T) monotonically rises in 1 / 3,3, so the minimum value of G (T) is g (1 / 3)
RELATED INFORMATIONS
- 1. The function f (x) = (1 / 3) ^ x, X belongs to [- 1,1], the minimum value of function g (x) = f (x) ^ 2-2af (x) + 3 is h (a), find H (a)?
- 2. The function f (x) = (1 / 3) ^ x, X ∈ [- 1,1]; the minimum value of function g (x) = f ^ 2 (x) - 2AF (x) + 3 is h (a) (1) Find H (a). (2) whether there are real numbers m, N, and meet the following conditions: ① m > n > 3; ② when the definition field of H (a) is [n, M], the value range is [n ^ 2, m ^ 2]. If there is, find the value of M, N; if not, explain the reason
- 3. Given the function f (x) = (1 / 3) ^ x, X belongs to [- 1,1], the minimum value of function g (x) = [f (x)] ^ 2-2af (x) + 3 is h (a) (1) When a ＞ = 1, find H (a) (2) Whether there is a real number m, while satisfying 1: M > n > 3,2: when the definition field of H (a) [n, M], whether the value field [n ^ 2, m ^ 2] exists, if it exists, find out the value of M, N, if it does not exist, explain the reason
- 4. Given the x power of function f (x) = (1 / 3), X belongs to [- 1,1], the minimum value of function g (x) = f ^ 2 (x) - 2AF (x) + 3 is h (a) (1) When a = 1, find the range of G (x) (2) The analytic expression of H (a)
- 5. It is known that the function f (x) = 1 / 3 to the power of X, X belongs to [- 1,1], and the minimum value of function g (x) = f (x) 2-2af (x) + 3 is h (a) For H (a), I calculate three expressions in three cases, but many of my classmates calculate - 6. Why?
- 6. Given that the image of the function f (x) = a ^ x + m passes through the point (1,7), and the image of his inverse function y = f ^ (- 1) passes through the point (4,0), find the value of a, M
- 7. Given the function f (x) = LNX ax. (1) when a > 0, judge the monotonicity of F (x) in the domain of definition; (2) if the minimum value of F (x) in [1, e] is 32, find the value of A
- 8. Given the function f (x) = log2 (1 + x) / (1-x), prove f (x1) + F (x2) = f ((x1 + x2) / (1 + x1x2))
- 9. Let f (x) = Log1 / 2 (10 ax), where a is a constant, f (3) = - 2, KKK Let f (x) = Log1 / 2 (10 ax), where a is a constant and f (3) = - 2 (1) The value of a (2) If for any x ∈ [3,4], the inequality f (x) ＞ (1 / 2) ^ x + m, find the value range of real number M
- 10. Given that the function f (x) = log (1-mx) / (x-1) is an odd function in the domain of definition, find m and its domain of definition
- 11. It is known that the function f (x) = 1 + √ (2x-3) has an inverse function, and the point m (a, b) is not only on f (x) but also on the inverse function f (x)
- 12. What is the inverse function of the function y = 1 + m (x-1) (x > 1)
- 13. If the inverse function image of F (x) = logm (x) passes (2, n), then the minimum value of N-M is
- 14. If f (x) = log3 (3x + 1), then the inverse function image passes through the point (x.y)?
- 15. How to find the inverse function of function y = 3x / (3x + 1)? Why is the answer y = log3 [x / (1-x)],
- 16. It is known that the function y = f (x) is an odd function. When x > 0, f (x) = log2x, then the value of F (f (1 / 16)) is equal to?
- 17. F (log2x) = X-1 / X to find f (x) It's the logarithm of X with base 2
- 18. Ask f (x) = 1 + F (1 / x) * log2x to find f (2) =? Log2x is superscript
- 19. Let f (x) = 2 + log2x (1 ≤ x ≤ 4) find the range of (1) y = [f (x)] ² + F (2x); (2) y = f (X & #178;) + F (2x)
- 20. If f (log2x) = x, then f (1 / 2)=