每日导数59

换主元，常用放缩与有界放缩

已知函数\(f(x)=a^2e^x-3ax+2\sin x-1\)

(1)若\(f(0)\)是函数\(f(x)\)的极值，求实数\(a\)的值

(2)证明，当\(a\geq 1\)时，\(f(x)\geq 0\)

解

(1)\(f^{\prime}(x)=a^2e^x-3a+2\cos x,f^{\prime}(0)=a^2-3a+2=0\)

得\(a=1\)或\(a=2\)

(2) 考虑\(\varphi(a)=a^2e^x-3ax+2\sin x-1\)

则\(\varphi^{\prime}(a)=2ae^x-3x\)，关于\(a\)单调递增

\(\varphi^{\prime}(a)=0\)得\(a=\dfrac{3x}{2e^x}\leq \dfrac{3x}{2ex}\leq \dfrac{3}{2e}<1\)

从而\(\varphi(a)\)在\(a\geq 1\)上单调递增

则\(\varphi(a)\geq \varphi(1)=e^x-3x+2\sin x-1\)

记\(\gamma(x)=e^x-3x+2\sin x-1\)

\(\gamma^{\prime}(x)=e^x+2\cos x-3\)

当\(x<0\)时，\(e^x+2\cos x-3<1-3+2=-2+2=0\)

当\(x\in\left(0,\dfrac{\pi}{2}\right)\)时，\(\gamma^{\prime}(x)=e^x+2\cos x-3>1+x+\dfrac{x^2}{2}+2\left(1-\dfrac{x^2}{2}\right)-3=x-\dfrac{x^2}{2}>0\)

当\(x\in\left(\pi,+\infty\right)\)时，\(\gamma^{\prime}(x)=e^x+2\cos x-3>e^{\pi}-5>e^{\pi}-5>0\)

当\(x\in\left[\dfrac{\pi}{2},\pi\right]\)时,\(\gamma^{\prime\prime}(x)=e^x-2\sin x>e^{\frac{\pi}{2}}-2>0\)

所以\(\gamma^{\prime}(x)\geq \gamma^{\prime}\left(\dfrac{\pi}{2}\right)=e^{\frac{\pi}{2}}-3>0\)

从而当\(x>0,\gamma^{\prime}(x)>0\)

而\(\gamma^{\prime}(0)=0\)

则\(\gamma(x)\geq \gamma(0)=0\)

综上\(f(x)=\varphi(a)\geq \varphi(1)=e^x-3x+2\sin x-1\geq \gamma(0)=0\)