# 2015年考研数二第10题解析

## 解析

$$f^{‘}(x) =$$

$$(x^{2}2^{x})^{‘} =$$

$$2×2^{x} + x^{2}2^{x} \ln 2.$$

$$f^{‘}(0) = 0+0=0.$$

$$f^{”}(x) =$$

$$(f^{‘}(x))^{‘} =$$

$$(2×2^{x} + x^{2}2^{x} \ln 2)^{‘} =$$

$$(2×2^{x})^{‘} + (x^{2}2^{x} \ln 2)^{‘} =$$

$$2 \cdot 2^{x} + 2x 2^{x} \ln 2 +$$

$$\ln 2 \cdot 2x 2^{x} + \ln 2 x^{2} 2^{x} \ln 2.$$

$$f^{”}(0) =2 \cdot 1 + 0 + 0 + 0 = 2.$$

$$f^{”’}(x) =$$

$$(f^{”}(x))^{‘} =$$

$$(2 \cdot 2^{x})^{‘} + (2x 2^{x} \ln 2)^{‘} +$$

$$(\ln 2 \cdot 2x 2^{x})^{‘} + (\ln 2 x^{2} 2^{x} \ln 2)^{‘} =$$

$$2 \cdot 2^{x} \ln 2 +$$

$$2 \ln 2 \cdot 2^{x} +$$

$$2 \ln 2 \cdot x 2^{x} \ln 2 +$$

$$2 \ln 2 \cdot 2^{x} +$$

$$2 \ln 2 \cdot x 2^{x} \ln 2 +$$

$$\ln 2 \cdot \ln 2 \cdot 2x 2^{x} +$$

$$\ln 2 \ln 2 \cdot x^{2} 2^{x} \ln 2.$$

$$f^{”’}(0) =$$

$$2 \ln 2 + 2 \ln 2 + 0 + 2 \ln 2 + 0 + 0 + 0 =$$

$$2 \times 3 \times \ln 2.$$

$$f^{””}(0) =$$

$$(f^{”’}(0))^{‘} =$$

$$(2 \cdot 2^{x} \ln 2)^{‘} +$$

$$(2 \ln 2 \cdot 2^{x})^{‘} +$$

$$(2 \ln 2 \cdot x 2^{x} \ln 2)^{‘} +$$

$$(2 \ln 2 \cdot 2^{x})^{‘} +$$

$$(2 \ln 2 \cdot x 2^{x} \ln 2)^{‘} +$$

$$(\ln 2 \cdot \ln 2 \cdot 2x 2^{x})^{‘} +$$

$$(\ln 2 \ln 2 \cdot x^{2} 2^{x} \ln 2)^{‘} \Rightarrow$$

$$具体求导过程省略 \Rightarrow$$

$$f^{””}(0) = 2 \times (\ln 2)^{2} \times 6.$$

$$f^{‘}(0) = 0;$$

$$f^{”}(0) = 2;$$

$$f^{”’}(0) = 2 \times 3 \times \ln 2;$$

$$f^{””}(0) = 2 \times 6 \times (\ln 2)^{2} \Rightarrow$$

$$f^{””}(0) = 3 \times 4 \times (\ln 2)^{2}.$$

$$f^{‘}(0) = (1)(1-1) (\ln 2)^{1-2};$$

$$f^{”}(0) = (2)(2-1)(\ln 2)^{2-2};$$

$$f^{”’}(0) = (3)(3-1)(\ln 2)^{3-2};$$

$$f^{””}(0) = (4)(4-1)(\ln 2)^{4-2}.$$

$$f^{(n)}(0) = n(n-1) (\ln 2)^{n-2}.$$

$$(x^{2}2^{x})^{(n)} =$$

$$C_{n}^{0}(x^{2})^{(0)}(2^{x})^{(n)} +$$

$$C_{n}^{1}(x^{2})^{(1)}(2^{x})^{(n-1)} +$$

$$C_{n}^{2}(x^{2})^{(2)}(2^{x})^{(n-2)} +$$



$$C_{n}^{n}(x^{2})^{(n)}(2^{x})^{(0)}.$$

$$(x^{2})^{(0)} = x^{2};$$

$$(x^{2})^{(1)} = 2x;$$

$$(x^{2})^{(2)} = 2;$$

$$(x^{2})^{(3)} = 0.$$

$$(x^{2}2^{x})^{(n)} =$$

$$C_{n}^{0}(x^{2})^{(0)}(2^{x})^{(n)} +$$

$$C_{n}^{1}(x^{2})^{(1)}(2^{x})^{(n-1)} +$$

$$C_{n}^{2}(x^{2})^{(2)}(2^{x})^{(n-2)} =$$

$$C_{n}^{0}(x^{2})(2^{x})^{(n)} +$$

$$C_{n}^{1}(2x)(2^{x})^{(n-1)} +$$

$$C_{n}^{2}(2)(2^{x})^{(n-2)}.$$

$$(x^{2}2^{x})^{(n)} =$$

$$C_{n}^{2}(2)(2^{x})^{(n-2)}.$$

$$(2^{x})^{(1)} = 2^{x} \ln 2;$$

$$(2^{x})^{(2)} = 2^{x} (\ln 2)^{2};$$

$$(2^{x})^{(3)} = 2^{x} (\ln 2)^{(3)}.$$

$$(2^{x})^{(n-2)} = 2^{x} (\ln 2)^{(n-2)}.$$

$$C_{n}^{2}(2)(2^{x})^{(n-2)} =$$

$$\frac{n!}{(n-2)! \cdot 2!} \cdot 2 \cdot 2^{x}(\ln 2)^{(n-2)} \Rightarrow$$

$$x=0 \Rightarrow$$

$$n(n-1)(\ln 2)^{(n-2)}.$$

EOF