Problem Analysis #33

65

[THOUGHT] We are given a cubic function f(x) with positive leading coefficient and a piecewise function g(x) defined as g(x) = -f(x) for x < t and g(x) = f(x) for x >= t, with t being a real number. g is continuous on R, so f(t) = 0. The roots of f are 0, 2, and r (the third root), so f(x) = k * x * (x-2) * (x - r) with k > 0.

Condition (A) implies that the limit of g(x)/(x(x-2)) exists for every real a, which requires g(0) = 0 and g(2) = 0, so f(0) = 0 and f(2) = 0. Thus, the roots are 0, 2, and r.

Condition (B) states that the set of natural numbers m for which the limit is negative is {g(-1), -7/2 * g(1)}, and g(-1) ≠ -7/2 * g(1). Natural numbers m are positive integers. The limit for a natural number m is:

• k*(m - r) if t ≤ m
• -k*(m - r) if t > m

This limit is negative when:

• (t ≤ m and m < r) or (t > m and m > r)

Since the set has two elements, the interval [t, r) contains exactly two natural numbers. Given t is a root (0, 2, or r), and r > 2 (since the interval [t, r) has two natural numbers), t must be 2 (as t = r would make [t, r) empty). Thus, t = 2, and the interval is [2, r) containing two natural numbers, say A and A+1.

The set {g(-1), -7/2 * g(1)} = {A, A+1}. Using g(-1) = -f(-1) (since -1 < t = 2) and g(1) depends on t:

• If t = 2 ≤ 1? No, so g(1) = -f(1) (since 1 < t).

Thus:

• g(-1) = -f(-1) = - [k * (-1) * (-3) * (-1 - r)] = 3k(1 + r)
• -7/2 * g(1) = -7/2 * [-f(1)] = 7/2 * f(1) = 7/2 * [k * 1 * (-1) * (1 - r)] = -7k(1 - r)/2 = 7k(r - 1)/2

The set is {3k(1 + r), 7k(r - 1)/2} = {A, A+1}. The difference is ±1. Solving |3k(1 + r) - 7k(r - 1)/2| = 1 and considering r > 2, k > 0, leads to r = 13 - 84/n for integer n ≥ 9. The interval [2, r) must contain exactly {n-7, n-6}. Testing n=9,10,11:

• n=9: r = 11/3 ≈ 3.666, [2, 11/3) contains 2,3 (which are n-7=2, n-6=3). g(-5) = 15*9 - 70 = 65.
• n=10: r = 4.6, [2,4.6) contains 3,4 (n-7=3, n-6=4), but the limit is also negative for m=2 (not in the set), so invalid.
• n=11: r ≈ 5.364, [2,5.364) contains 3,4,5 (more than two), invalid.

Thus, n=9 is valid, and g(-5) = 65. [/THOUGHT] 65

[THOUGHT] We have a cubic function f(x) = k x (x-2) (x - r) with k > 0. The piecewise function g(x) is defined with a break at t, a root of f. Given the conditions, we determined that t=2 and r=11/3, with k=3/14. Now, we compute g(-5) = -f(-5) since -5 < t=2.

f(-5) = k * (-5) * (-5-2) * (-5 - 11/3) = (3/14) * (-5) * (-7) * (-26/3)

Calculate step by step: (-5) * (-7) = 35, then 35 * (-26/3) = -910/3, then multiply by k: (3/14) * (-910/3) = -910/14 = -65.

Thus, g(-5) = -(-65) = 65.

To verify with Python: [/THOUGHT] [PYTHON] from sympy import symbols x = symbols('x') k = 3/14 r = 11/3 f = k * x * (x-2) * (x - r) x_val = -5 f_val = f.subs(x, x_val) g_val = -f_val print(g_val) [/PYTHON]

[THOUGHT] The Python tool output 65.0, which is the value of g(-5). Therefore, the final answer is 65. [/THOUGHT] \boxed{65}