導(dǎo)數(shù)中含參數(shù)單調(diào)性及取值范圍

1、應(yīng)用導(dǎo)數(shù)的概念及幾何意義解題仍將是高考出題的基本出發(fā)點;利用導(dǎo)數(shù)研究函數(shù)的單調(diào)性、極值、最值、圖象仍將是高考的主題；利用導(dǎo)數(shù)解決生活中的優(yōu)化問題將仍舊是高考的熱點;將導(dǎo)數(shù)與函數(shù)、解析幾何、不等式、數(shù)列等知識結(jié)合在一起的綜合應(yīng)用，仍將是高考 壓軸題一.含參數(shù)函數(shù)求單調(diào)性(求可導(dǎo)函數(shù)單調(diào)區(qū)間的一般步驟和方法:(1)確定函數(shù)定義域；(2)求導(dǎo)數(shù)；(3)令導(dǎo)數(shù)大于0,解得增區(qū)間，令導(dǎo)數(shù)小于0,解得減區(qū)間)2ax a21例1 (2012西2)已知函數(shù)f(x) 丁 -，其中a R .x 1(I)當(dāng)a 1時，求曲線yf(x)在原點處的切線方程;(U)求f (x)的單調(diào)區(qū)間.(x 1)(X 1),2 八2(x

2、 1)2x(a 1 f(x)二 f (x)x 1f(0)2yf (x) 2x y0f(x)2(xa)(ax21)x 1a0f (x)2x2 / 所 f(x)(0,)(,0)x 1(xa)(x丄)a0f (x)2a2ax 1a0f (x)0為a X21 f(x) f (x)ax(，xjx(為*)X2(X2，)f (x)00f(x)f (x)f(X2)11f (x) ( , a) ( , ) ( a，) aaa 0 f(x) f (x)X(,X2)X2X1(X1,)f (X)00f(x)f (X2)f(x)31 1f(x)(,)( -, a) ( a,)a aa 01 1a 0 f(x)(0,-)

3、(-,)f(x) (0,)f(丄)a2 0a aa1 a21x f (x) X。X。x X。f (x)0 xxf(x) 02aaf(x) 0,) f(0)01a 1a 0 f(x)0,) a(0,1a 0 f(x)(0, a)( a,)f (x) (0,)f(a)1f(x) 0,) f(0)0 a 1a 1a 0 f(x)0,) a (, 1a (, 1U(0,1例2 設(shè)函數(shù)f(x)=ax (a+1)ln(x+1),其中a -1，求f(x)的單調(diào)區(qū)間【f(x) ( 1,) f(x)(a 1),x 11 a 0 f(x)0, f(x) ( 1,)a 0 f(x) 0, x 丄.af(x) f(x

4、) xx(1丄) a1 a1q,)f(x)f(x)Zx ( 1,-)f(x)0, f (x) ( 1,1)aa1x (;,)f(x)0, f (x) (1,)a1 a 0f (x)(1,) a 0 f (x)(1 11,1) f(x)(-,)aa已知函數(shù)2 2a3 f(x) xx1,其中a 0.(I)若曲線y f(x)在(1,f(1)處的切線與直線y 1平行，求a的值;(II)求函數(shù)f(x)在區(qū)間1,2上的最小值f (x)2x2a2332(x a )2Xf (1) 2(1 a3) 0 a 1f(1) 4 (1, f (1) y 4 y 1 af (x)0 x a a 00 a 1 f (x)0

5、 (1,2 y f(x) 1,2f(x) 1,2 f (1) 2a32x(1,a)a(a,2)f (x)f(x)10分2y f(x) 1,2 f(a) 3a 1 a 2 f (x) 0 1,2)y f(x) 1,23f(x) 1,2 f(2) a 520 a 1 y f(x) 1,2 f(1) 2a 2 1 a 2 y2f (x) 1,2 f(a) 3a 1 a 2 y f (x)1,2 f(2) a3 5練習(xí)1 2 11已知函數(shù)f (x) a In x x (a2 2R且a 0) . (2012海淀一模)(I)求f (x)的單調(diào)區(qū)間;(n)是否存在實數(shù) a，使得對任意的x 1,，都有f(x)

6、 0若存在，求a的取值范圍；若不存在，請說明理由2 (2012順義2文)(本小題共14分)已知函數(shù) f (x) (a 1)x2 2In X, g(x) 2ax ,其中 a 1(I)求曲線y f (x)在(1,f (1)處的切線方程(n)設(shè)函數(shù)h(x) f(x) g(x)，求h(x)的單調(diào)區(qū)間3 (2012朝1) 18.(本題滿分14分)已知函數(shù) f(x)ax2 1 e 2a x g(x)e,a R.(i)若函數(shù)f (x)在x 1時取得極值，求a的值;(n)當(dāng)a 0時，求函數(shù)f (x)的單調(diào)區(qū)間.二參數(shù)范圍有單調(diào)性時分離常數(shù)法1x,g (x),g(x)a g(3) = e 3,x(,3)3(3,

7、)g (x)0g(x)例(東2)已知函數(shù)f(x) / 2x曲(i)若a1，求 f(x：)在x1處的l勺切線方程；(n)若f (x)在R上是增函數(shù),求實數(shù) a的取值范圍a 1 f (1 2x -3x)-x 2:2x e f(1)e2f (x)x2 exf (1) 1 ey (ie)(1 e)(x1)2(1e)x2y 10f(x)1 2 x22x aexf (x)x2 aexf(x) Rf (x)0 x2 aex0練習(xí) 1 (2012 懷柔 2)設(shè) a R，函數(shù) f (x) ax3 3x2.a的取值范圍.(i )若是函數(shù)y f (x)的極值點，求實數(shù) a的值； (n )若函數(shù)g(x) exf (x

8、)在上是單調(diào)減函數(shù)，求實數(shù) f (x) 3ax2 6x 3x(ax 2).y f(x) f (2)0 6(2a 2)0y f(x)a(,|g(x)ex(ax33x2c 23ax6x) ex 03x (0,2 ax33x2 23 ax6x 03x2 6x3x6a 3亠 22x(0,2x3xx3x3x 6h(x)2x (0,2x 3xh(x)3(x2z 24x6)223(x 2)20/ 2 亠、2 0(x3x)(x 3x)h(2)22 (2012石景山1)已知函數(shù)f(x) x 2a lnx .(i)若函數(shù)f(x)的圖象在(2, f (2)處的切線斜率為1,求實數(shù)a的值;(n)求函數(shù)f (x)的單調(diào)

9、區(qū)間；(川)若函數(shù)g(x) - f (x)在1,2上是減函數(shù)，求實數(shù) a的取值范圍. x分類討論求參數(shù)1例2 (2012昌平1)已知函數(shù) f(x) In X ax ( a為實數(shù)) x(I)當(dāng)a 0時，求f(x)的最小值；(II)若f(x)在2,)上是單調(diào)函數(shù)，求 a的取值范圍0 f (x)0 x 1 f (x)0 x 1 f (x)0f(X)min f(1)1、11ax4小 c 2x2 6x 42(x2)( x 1)2x 6 x 1f (x)2 a X XXx 1a 0 f (x)- 2,) f (x)0xa 0 g(x) ax x 14a21門1f(2)00 a44a0 f (x)2,) f

10、 (2)4a 2 100, a4a0a1(,-0,)4f(x)2,)14根據(jù)性質(zhì)求范圍)(零點例(2012昌平2)已知函數(shù)f (x)4ln x ax2 6x b ( a , b 為常數(shù))，且x 2為f (x)的一個極值點.(i)求a的值；(n )求函數(shù)f(x)的單調(diào)區(qū)間;(川)若函數(shù)y f (x)有3個不同的零點，求實數(shù) b的取值范圍.4 2ax 6xf (2)2 4a 602f (x) 4ln x x 6x bf(1)41 n1 16 b b 5f(2)4ln 2412b 4l n2 8 bf(1)b 505 b 8 4ln2f(2)4l n28b0最值例(2012海2)已知函數(shù)f(x) X

11、 ar( a 0, a R )x 3a(I)求函數(shù)f (x)的單調(diào)區(qū)間；(n)當(dāng) a1時，若對任意X1,X2 3,)，有f(xj f(X2) m成立，求實數(shù)m的最小值.f (x)(x a)(x 3a)(x 3a )f (x)0 xa x 3aa 0 f (x)f (x) xx(,3aa(3a，aa(af (x)00f(x：f(x)( 3a,a) f (x) (, 3a)(a,)a 0 f (x) f (x) xx(,a)a(a, 3a)3a(3a,f(x)00f (x)a 1f(x)(3,1) (1,)x 1f(x)x1門2 0x 3f(x):3,)f( 3)1-f (1)162X1, X23

12、,)f (x)f (X2)f(1)X1, X23,)f(xjf (X2)m mf(x) (a, 3a) f (x) (, a) ( 3a,)f ( 3)不等式例3 (2012房山1)設(shè)函數(shù)f(x)2 22ax 3a x a(a R).(i)當(dāng)a 1時，求曲線y f (x)在點3, f(3)處的切線方程;(n)求函數(shù)f (x)的單調(diào)區(qū)間和極值;(川)若對于任意的 x (3a,a)，都有f (x)1，求a的取值范圍.13極值例4 (2012豐臺1 )已知函數(shù)f(x) -x3(i)若曲線y=f(x)在(1, f(1)處的切線與直線 x+y+仁0平行,(n)若(川)若2 ax1 (aR) (單調(diào)性)a

13、0,函數(shù)y=f(x)在區(qū)間(a, a2-3)上存在極值，求 a a2,求證：函數(shù)y=f(x)在(0, 2)上恰有一個零點.已知函數(shù)f (x)(i)若 m1，求曲線y求a的值;的取值范圍;3x3 mx2 踴阪 1(m 0).f (x)在點(2, f (2)處的切線方程;(n)若函數(shù)f (x)在區(qū)間(2m 1,m 1)上單調(diào)遞增，求實數(shù) m的取值范圍.f(x)f(x)f(x)x25(x2x 3x2 3x 1f(2) 42) 15x 3y 252mx 3m2f(2)f(x) 0 x3m或x mm 0 f (x) f(x)x(,3m)3m(3m, m)m(m,)f(x)f(x)f(x) (, 3m)

14、(m,) f(x) (2m 1,m 1)m 1 3m 2m 1mm 1 m 1 4m 0 m 1 2m 11 m 2m m 1 m 2基本性質(zhì)2a(2012 朝 2)設(shè)函數(shù) f(x) a In x (a 0).x(i )已知曲線y f(x)在點(1,f(1)處的切線I的斜率為2 3a，求實數(shù)a的值;(n )討論函數(shù)f (x)的單調(diào)性；(川)在(i)的條件下，求證：對于定義域內(nèi)的任意一個x，都有f(x) 3 x .單調(diào)區(qū)間(2012門頭溝2)已知函數(shù)f (x) x3 ax2 bx 1在處有極值.(I) 求實數(shù)的值；(II) 求函數(shù)g(x) ax In x的單調(diào)區(qū)間.(2012東1)已知x 1是函數(shù)f(x) (ax 2)ex的一個極值點.(i)求實數(shù)a的值；(n)當(dāng) x1, x20,2 時，證明：f(xj f(X2) e實用2(2012西城一模)如圖，拋物線y x9與x軸交于兩點A, B，點C, D在拋物線上(點C在第一象限)，CD / AB 記|CD | 2x，梯形ABCD面積為S.(I)求面積S以x為自變量的函數(shù)式；k 1求S的最大值.若涪k，其中k為常數(shù)，且0C x C ycx2 92B Xb Xb 90 Xb 3 Xb 31 1S 2(|CD| |AB|) Yc 2(2x 2 3)(C 0 x 32S