📝 题目
例 4 设 $f\left( x\right)$ 在 $\mathbf{R}$ 上连续可导,且 $\mathop{\sup }\limits_{{x \in \mathbf{R}}}\left| {{\mathrm{e}}^{-{x}^{2}}{f}^{\prime }\left( x\right) }\right| < + \infty$ ,求证:
$$ \mathop{\sup }\limits_{{x \in \mathbb{R}}}\left| {x{\mathrm{e}}^{-{x}^{2}}f\left( x\right) }\right| < + \infty . $$
💡 答案与解析
证法 1 设 $M = \mathop{\sup }\limits_{{x \in \mathbf{R}}}\left| {{\mathrm{e}}^{-{x}^{2}}{f}^{\prime }\left( x\right) }\right|$ ,则 $\left| {{f}^{\prime }\left( x\right) }\right| \leq M{\mathrm{e}}^{{x}^{2}}$ ,且
$$ \left| {f\left( x\right) }\right| \leq \left| {f\left( x\right) - f\left( 0\right) }\right| + \left| {f\left( 0\right) }\right| \leq \left| {{\int }_{0}^{x}\left| {{f}^{\prime }\left( t\right) }\right| \mathrm{d}t}\right| + \left| {f\left( 0\right) }\right| $$
$$ \leq M\left| {{\int }_{0}^{x}{\mathrm{e}}^{{t}^{2}}\mathrm{\;d}t}\right| + \left| {f\left( 0\right) }\right| \;\left( {\forall x \in \mathbf{R}}\right) . \tag{7.16} $$
根据