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(2013年常州)已知函数f(x)=x|x-a|-lnx.
(1)若a=1,求函数f(x)在区间[1,e]的最大值;(2)求函数f(x)的单调区间;(3)若f(x)>0恒成立,求a的取值范围.
这是一道期末调研压轴题,着重考查学生分类讨论的运用和计算能力,第(1)问此处不再累述,第(2)问答案如下.
当a<1时,f(x)单调递减区间是(0,(a+√a2+8)/4),f(x)单调递增区间是((a+√a2+8)/4,+∞);
当1≤a≤2√2时,f(x)单调递减区间是(0,a),f(x)单调的递增区间是(a,+∞); 相似文献
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<正>问题背景在讨论函数零点个数时,一般采用研究函数的单调性,结合零点存在性定理进行严密地论证.例如,当我们论证出f(x)在区间(a,x0)上单调递减,在(x0,b)上单调递增,且f(x0)<0时,为了严密论证f(x)在(a,b)上有两个零点,需在x0左侧取出f(x1)>0,右侧取出f(x2)>0,才能得出f(x)共有两个零点的结论,这类问题一般称之为取点问题,在高考真题中十分常见. 相似文献
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本文主要研究非自治时标动力学方程x△=f(t,x),(t,x)∈T×Rn的平凡解稳定、一致稳定、渐近稳定,与不稳定的充分必要条件.在定理的证明中充分性主要利用K类函数严格单调递增和Lyapunov函数的连续性;必要性的证明中,分别构造Lyapunov函数,满足定理的条件,使得定理的结论成立. 相似文献
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《数学通报》2010年4月第1848号数学问题为:
已知函数:f(x)=x3+bx,数列{an},其中a1>0.
(1)若an=f(n),当数列{an}为递增数列时,求b的取值范围;
(2)若an+1=f(an),当数列{an}为递增数列 时,求首项a1的取值范围.(用b表示,且b≥0)
原解答对于(1),将数列{an=f(n)}递增数列转化为函数f(x) =x3 +bx在[1,+∞)单调递增,进而转化为f′(x) =3x2+b≥0在[1,+∞)上恒成立,从而求出b的范围是:b≥-3. 相似文献
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<正>1引言函数的单调性和奇偶性是函数的基本性质.常见的函数单调性的求法有:(1)定义法;(2)图象法;(3)导数法.还有一些与函数单调性有关的结论:若函数f(x),g(x)均为增(减)函数,则f(x)+g(x)为增(减)函数;若f(x)为增(减)函数,则-f(x)为增(减)函数;若函数f(x)为增(减)函数且f(x)>0, 相似文献
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设函数 f (x)在 (-∞ , ∞ )上连续 ,当 x≠ 0时 ,我们称 F(x) =1x∫x0 f (t) dt为 f (x)在 [0 ,x]上的平均值函数 ,本文将介绍平均值函数 F(x)的若干性质并举例说明其应用 .一、F(x)的性质性质 1 f(x)是 [0 ,x](或 [x,0 ])上的有界函数 ,F(x)也是 [0 ,x]或 [x,0 ]上的有界函数 .性质 2 若 f (x)为奇 (偶 )函数 ,则 F(x)也为奇 (偶 )函数 .性质 3 若 f(x)是周期为 T(T>0 )的周期函数 ,则limx→ ∞1x∫x0f (t) dt=1T∫T0f (t) dt (1 ) 性质 4 若 f(x)为单调递增 (减 )函数 ,则 F(x)也为单调递增 (减 )函数 .性质 5 若对任意… 相似文献
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R. R. Salimov 《Siberian Mathematical Journal》2012,53(4):739-747
Under study is the class of ring Q-homeomorphisms with respect to the p-module. We establish a criterion for a function to belong to the class and solve a problem that stems from M. A. Lavrentiev [1] on the estimation of the measure of the image of the ball under these mappings. We also address the asymptotic behavior of these mappings at a point. 相似文献
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F. J. Schuurmann P. R. Krishnaiah A. K. Chattopadhyay 《Journal of multivariate analysis》1973,3(4):445-453
In this paper, the authors cosider the derivation of the exact distributions of the ratios of the extreme roots to the trace of the Wishart matrix. Also, exact percentage points of these distributions are given and their applications are discussed. 相似文献
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Michael Coons 《The Ramanujan Journal》2013,30(1):39-65
Let $\mathcal{G}(z):=\sum_{n\geqslant0} z^{2^{n}}(1-z^{2^{n}})^{-1}$ denote the generating function of the ruler function, and $\mathcal {F}(z):=\sum_{n\geqslant} z^{2^{n}}(1+z^{2^{n}})^{-1}$ ; note that the special value $\mathcal{F}(1/2)$ is the sum of the reciprocals of the Fermat numbers $F_{n}:=2^{2^{n}}+1$ . The functions $\mathcal{F}(z)$ and $\mathcal{G}(z)$ as well as their special values have been studied by Mahler, Golomb, Schwarz, and Duverney; it is known that the numbers $\mathcal {F}(\alpha)$ and $\mathcal{G}(\alpha)$ are transcendental for all algebraic numbers α which satisfy 0<α<1. For a sequence u, denote the Hankel matrix $H_{n}^{p}(\mathbf {u}):=(u({p+i+j-2}))_{1\leqslant i,j\leqslant n}$ . Let α be a real number. The irrationality exponent μ(α) is defined as the supremum of the set of real numbers μ such that the inequality |α?p/q|<q ?μ has infinitely many solutions (p,q)∈?×?. In this paper, we first prove that the determinants of $H_{n}^{1}(\mathbf {g})$ and $H_{n}^{1}(\mathbf{f})$ are nonzero for every n?1. We then use this result to prove that for b?2 the irrationality exponents $\mu(\mathcal{F}(1/b))$ and $\mu(\mathcal{G}(1/b))$ are equal to 2; in particular, the irrationality exponent of the sum of the reciprocals of the Fermat numbers is 2. 相似文献
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N. K. Bakirov 《Journal of Mathematical Sciences》1989,44(4):425-432
One investigates the asymptotic properties of the quantile test, similar to the properties of the Pearson's chi-square test of fit.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 153, pp. 5–15, 1986.The author is grateful to D. M. Chibisov for useful remarks. 相似文献
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