一、选择题详解
题目1：已知函数\(f(x)=x^3-3x^2+4x+6\)，求\(f(x)\)的极值。
解题思路：首先求出函数的导数，然后令导数等于0，求出驻点，再通过导数的符号变化确定极值。
解题步骤：
- 求导数：\(f'(x)=3x^2-6x+4\)。
- 令\(f'(x)=0\)，解得\(x_1=1\)，\(x_2=\frac{2}{3}\)。
- 检查驻点两侧导数的符号，确定极值。
答案：\(f(x)\)在\(x=1\)处取得极大值\(f(1)=8\)，在\(x=\frac{2}{3}\)处取得极小值\(f\left(\frac{2}{3}\right)=\frac{58}{27}\)。
二、填空题详解
题目2：若等差数列\(\{a_n\}\)的前\(n\)项和为\(S_n\)，且\(S_5=55\)，\(S_8=165\)，则$a_6= ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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