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o3mh는 2017 수능 30을 제외하면 전부 오답

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2017 수능 30 스펙터, 팬텀, 네뷸라 정답

A function f(x) is defined for x > a, and a quartic polynomial g(x) with a leading coefficient of -1 satisfies the following conditions, where a is a constant:

For all real numbers x > a, (x - a)f(x) = g(x).

For two distinct real numbers b and c, the function f(x) attains the same local maximum value M at x = b and x = c, where M > 0.

The number of local extrema of f(x) is greater than the number of local extrema of g(x).

Given that c - b = 6√3, find the minimum value of M. 216


2018 수능 30 스펙터, 팬텀, 네뷸라 정답

For a real number t, let the function f(x) be defined as f(x) = 1 - |x - t| if |x - t| ≤ 1, and f(x) = 0 if |x - t| > 1. 

For some odd integer k, let the function g(t) be defined as g(t) = integral from k to k+8 of f(x)cos(πx)dx. 

The function g(t) satisfies the following condition: 

Let alpha_1, alpha_2, ..., alpha_m (where m is a natural number) be all values of alpha where g(t) has a local minimum at t = alpha and g(alpha) < 0, 

listed in increasing order. Given that the sum from i=1 to m of alpha_i = 45. 

Find the value of k - π² multiplied by the sum from i=1 to m of g(alpha_i). 21


2019 수능 30 네뷸라 정답 / 팬텀,스펙터 오답

For a cubic function f(x) with a leading coefficient of 6π, the function g(x) is defined as:

g(x) = 1 / (2 + sin(f(x)))

g(x) has a local maximum or local minimum at x = α, and let α₁, α₂, α₃, α₄, α₅, ... be all α ≥ 0 where g(x) has a local extremum, 

listed in increasing order from smallest to largest. 

g(x) satisfies the following conditions:

(A) α₁ = 0 and g(α₁) = 2/5.

(B) 1 / g(α₅) = 1 / g(α₂) + 1/2

When g'(-1/2) = aπ, find the value of a².

(Given: 0 < f(0) < π/2) 27


2018 6월 30 네뷸라, 스펙터 정답 / 팬텀 오답

For a real number a and a function f(x) = ln(x⁴+1) - c (where c is a positive constant), let function g(x) be defined as:

g(x) = ∫_(a)^(x) f(t)dt

Let α₁, α₂, ..., α_m (where m is a natural number) be all values of a, 

listed in increasing order from smallest to largest, such that the graph of the function y = g(x) intersects the x-axis at exactly 2 distinct points.

When a = α₁, the function g(x) and constant k satisfy the following conditions:

(Condition (a)) Function g(x) has a local minimum at x = 1.

(Condition (b)) ∫_(α₁)^(α_m) *g(x)dx = kα_m ∫_(0)^(1) |f(x)|dx

Find the value of mk × e^c. 16


2024 수능 미적 30 스펙터, 네뷸라 정답 / 팬텀 오답

Let $f(x)$ be a differentiable function on the set of real numbers, 

and let its derivative $f'(x)$ be given by: $$f'(x) = |\sin x|\cos x$$

For a positive number $a$, let $y=g(x)$ be the equation of the tangent line to the curve $y=f(x)$ at the point $(a, f(a))$.  

Consider the function $$h(x) = \int_{0}^{x} \{f(t) - g(t)\} dt$$

When all positive numbers $a$ for which $h(x)$ has a local maximum or local minimum at $x=a$ are arranged in increasing order, let the $n$-th number be $a_n$.  

Find the value of $$\frac{100}{\pi} \times (a_6 - a_2)$$ 125


2023 6월 미적 30번 네뷸라, 스펙터, 팬텀 오답

For a positive number a, the function f(x) is given by:

f(x) = (x² - ax) / e^x

For a real number t, let g(t) be the number of distinct real roots of the equation in x:

f(x) = f'(t)(x - t) + f(t)

When g(5) + lim<sub>t→5</sub> g(t) = 5, and lim<sub>t→k⁻</sub> g(t) ≠ lim<sub>t→k⁺</sub> g(t) is satisfied,

the sum of all real values of k that satisfy the above condition is q/p.

Find the value of p + q. 

(where p and q are coprime natural numbers.) 16