三角幂函遇指函,分母指函求导易。
【例】已知函数\(f(x) = {e^x} – \sin x – \cos x\),\(g(x) = {e^x} + \sin x + \cos x\).
(1)证明:当\(x> – \dfrac{{5\pi }}{4}\)时,\(f(x) \ge 0\);
(2)若\(g(x) \ge 2 + ax\),求\(a\).
【解析】(1)令\(h(x) = \dfrac{{f(x)}}{{{e^x}}} = 1 – \dfrac{{\sin x + \cos x}}{{{e^x}}}\)(\(x> – \dfrac{{5\pi }}{4}\)),
则\(h'(x) = \dfrac{{2\sin x}}{{{e^x}}}\)
【问题1】什么时候一定有\({e^x}>\sin x + \cos x\)?
注意到当\(x> – \dfrac{{5\pi }}{4}\)时,\(\sin x + \cos x = \sqrt 2 \sin (x + \dfrac{\pi }{4}) \in [ – \sqrt 2 ,\sqrt 2 ]\),
\({e^{\dfrac{\pi }{2}}}>e>\sqrt 2 \),因此当\(x>\dfrac{\pi }{2}\)时,一定有\({e^x}>\sin x + \cos x\)。
于是可以分下面的情况讨论:
当\( – \dfrac{{5\pi }}{4}<x< – \pi \)时,\(\sin x>0\),\(h'(x)>0\),\(h(x)\)单调递增;
当\( – \pi <x<0\)时,\(\sin x<0\),\(h'(x)<0\),\(h(x)\)单调递减;
当\(0<x<\dfrac{\pi }{2}\)时,\(\sin x>0\),\(h'(x)>0\),\(h(x)\)单调递增.
①当\( – \dfrac{{5\pi }}{4} \le x \le \dfrac{\pi }{2}\)时,\(h{(x)_{\min }} = \min \{ h( – \dfrac{{5\pi }}{4}),h(0)\} \),
注意到\(h( – \dfrac{{5\pi }}{4}) = 1,h(0) = 0\)
从而当\( – \dfrac{{5\pi }}{4}<x \le \dfrac{\pi }{2}\)时,\(h(x) \ge 0\)
②当\(x>\dfrac{\pi }{2}\)时,\(\sin x + \cos x = \sqrt 2 \sin (x + \dfrac{\pi }{4}) \le \sqrt 2 \),\({e^x}>{e^{\dfrac{\pi }{2}}}>e>\sqrt 2 \),
故\(\dfrac{{\sin x + \cos x}}{{{e^x}}}<1\),\(\therefore h(x)>0\)
综上所述,当\(x> – \dfrac{{5\pi }}{4}\)时,\(h(x) \ge 0\),即\(f(x) \ge 0\)
(2)令\(\varphi (x) = 1 + \dfrac{{\sin x + \cos x – 2 – ax}}{{{e^x}}}\),则\( \varphi ‘(x) = \dfrac{{ – 2\sin x – a + 2 + ax}}{{{e^x}}} \)
注意到\(\varphi (0) = 0\),则\(\varphi (x) \ge 0 = \varphi (0)\)恒成立
故\(\varphi (x)\)在\(x = 0\)处取得最小值,由于\(\varphi (x)\)在\(R\)上连续可导,
故\(\varphi (x)\)在\(x = 0\)处取得极小值,从而\(\varphi ‘(0) = 0\),解得\(a = 2\)
另一方面,当\(a = 2\)时,\(\varphi {‘}(x) = \dfrac{{ – 2\sin x + 2x}}{{{e^x}}}\)
令\(u(x) = – 2\sin x + 2x\),则\(u'(x) = – 2\cos x + 2 \ge 0\)
\(\therefore u(x)\)在\(R\)上单调递增,注意到\(u(0) = 0\),
故当\(x<0\)时,\(u(x)<0\),\(\varphi ‘(x)<0\),\(\varphi (x)\)单调递减;
当\(x>0\)时,\(u(x)>0\),\(\varphi ‘(x)>0\),\(\varphi (x)\)单调递增.
\(\therefore \varphi (x) \ge \varphi (0) = 0\)恒成立
综上所述,\(a\)的值为\(2\)
原创文章,作者:leopold,如若转载,请注明出处:https://www.math211.com/2021/02/10/84/