7fed82726827c12d8b5b8f39ed25aeee1fda38b40696636153f410ac02692d525293e36dc839270ecc113558b84f06c11076be


이거를 Gemini1.5에게 영어로 바꿔달라고 해서


Absolutely! Here's the problem translated into English:

Problem 21:

For two real numbers a and b, the function f(x) is defined as follows:

f(x) = { 2^(x+a), if x ≤ 0 (x+b)^2, if x > 0 }

Let g(t) be the number of distinct real roots of the equation f(x) = t for a real number t. The function g(t) satisfies the following conditions:

There exists a positive real number k such that both

lim_(t→2k⁻) g(t) ≠ lim_(t→2k⁺) g(t) lim_(t→k⁻) g(t) ≠ lim_(t→k⁺) g(t)

hold.

For the ordered pair (a, b) of two real numbers that make the following equation true,

lim_(t→16⁻) g(t) × lim_(t→16⁺) g(t) = 2

find the product of the maximum and minimum values of a+b.

Translation Notes:

  • f(x) is a piecewise function: It has two different definitions for x less than or equal to 0 and for x greater than 0.
  • g(t) represents the number of roots: For a given value of t, g(t) tells us how many different values of x satisfy the equation f(x) = t.
  • Limits and discontinuities: The conditions on g(t) involve limits as t approaches certain values from the left and right, indicating that the function g(t) has discontinuities at these points.
  • Product of maximum and minimum: The final question asks for the product of the largest and smallest possible values of a+b, given the constraints on g(t).

Essentially, the problem is asking you to find the values of a and b that satisfy a set of conditions related to the function g(t), which is defined in terms of the piecewise function f(x).

Would you like me to help you solve this problem? If so, please let me know if you have any specific questions or if you'd like me to walk you through the solution step by step.


이렇게 하고 이걸 o1-preview한테 풀라고 했더니


대충푸는 척하면서(길어서 여기 붙여넣어지지가 않네)
Answer:

24−162

(24-16루트2)

이지랄 하는데 답은 -9 이거든

2024년9월4일 고2 문제라는데
아직 힘드나?


밑에는 인간(호랭이닷컴)의 해설과 정답이다



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