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已知函数\(f(x)=e^x-x-1\)
(1)讨论\(f(x)\)的单调性
(2)当\(x\geq 0,\)时\(f(2x)\geq 4x^3-ax^2\)，求\(a\)的取值范围
解
(1)略
(2)\(e^{2x}-2x-1\geq 4x^3-ax^2\)
当\(x=0\)，合题
当\(x\neq 0\)，\(a\geq -\dfrac{e^{2x}-4x^3-2x-1}{x^2}\)
\(g(x)=-\dfrac{e^{2x}-4x^3-2x-1}{x^2}\)
\(g^{\prime}(x)=-\dfrac{2(x-1)(e^{2x}-2x^2-2x-1)}{x^3}\)
因\(e^x\geq 1+x+\dfrac{x^2}{2}\)
则\(e^{2x}\geq 1+2x+2x\)
从而\(e^{2x}-2x^2-2x-1\geq 0\)
则不难得到\(g^{\prime}(x)\)在\((0,1)\)上正，在\((1,+\infty)\)上负
从而\(g(x)_{\max}=g(1)=7-e^2\)
则\(a\geq 7-e^2\)
标签：prime,geq,4x,导数,54,每日,2x,dfrac,ax From： https://www.cnblogs.com/manxinwu/p/18010382