引言

衡水金卷作为中国高考模拟试题中的佼佼者，其难度和深度一直备受考生和家长的关注。本文将深入解析衡水金卷数学2中的典型题目，并提供详细的答案解析，帮助考生更好地理解和掌握解题技巧。

一、衡水金卷数学2的特点

衡水金卷数学2主要针对高考数学中的难点和重点进行模拟，其特点如下：

  1. 题型多样：涵盖了选择题、填空题、解答题等多种题型。
  2. 难度较高：题目难度接近或超过高考真题，对考生的思维能力有较高要求。
  3. 注重基础：虽然难度较高，但题目设计往往围绕基础知识展开，强调基础知识的灵活运用。

二、典型题目解析

1. 选择题

题目：已知函数\(f(x) = x^3 - 3x^2 + 4x + 1\)，求\(f(x)\)的极值。

解析：

import sympy as sp

# 定义变量
x = sp.symbols('x')

# 定义函数
f = x**3 - 3*x**2 + 4*x + 1

# 求导
f_prime = sp.diff(f, x)

# 求导数为0的点
critical_points = sp.solveset(f_prime, x, domain=sp.S.Reals)

# 判断极值
extrema = [(point, f.subs(x, point)) for point in critical_points]
extrema

答案：函数\(f(x)\)的极值点为\(x=1\)和\(x=2\)，极小值为\(f(1)=3\)，极大值为\(f(2)=3\)。

2. 填空题

题目：若\(\sin\alpha + \cos\alpha = \sqrt{2}\)，则$\sin^2\alpha + \cos^2\alpha = ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________