Kho sỏt hm s Trn S Tựng Baứi Gii cỏc phng trỡnh sau: a) x -2 + 4- x = c) x + (1 - x )5 = b) x + x = x + Baứi Tỡm m cỏc phng trỡnh sau cú nghim: a) x + x + = m b) 16 - x + + x - (2 - x )(2 + x ) = m + x + - x - (3 + x )(6 - x ) = m d) - x + + x - (7 - x )(2 + x ) = m Baứi Tỡm m cỏc bt phng trỡnh sau nghim ỳng vi mi x ẻ R: c) a) x + x + > m b) m x + < x + m c) mx - x + m Baứi Cho bt phng trỡnh: x - x + x - + m < a) Tỡm m bt phng trỡnh cú nghim thuc [0; 2] b) Tỡm m bt phng trỡnh tho mi x thuc [0; 2] Baứi Tỡm m cỏc bt phng trỡnh sau: a) mx - x - Ê m + cú nghim b) (m + 2) x - m x + cú nghim x ẻ [0; 2] c) m( x - x + 1) Ê x + x + nghim ỳng vi mi x ẻ [0; 1] Trang 14 Trn S Tựng Kho sỏt hm s IV IM UN CA TH nh ngha: im U ( x0 ; f ( x0 ) ) gl im un ca th hm s y = f(x) nu tn ti mt khong (a; b) cha im x0 cho trờn mt hai khong (a; x0) v (x0; b) tip tuyn ca th ti im U nm phớa trờn th cũn trờn khong tip tuyn nm phớa di th Tớnh cht: ã Nu hm s y = f(x) cú o hm cp hai trờn mt khong cha im x0, fÂÂ(x0) = v fÂÂ(x) i du x i qua x0 thỡ U ( x0 ; f ( x0 ) ) l mt im un ca th hm s ã th ca hm s bc ba y = ax + bx + cx + d (a 0) luụn cú mt im un v ú l tõm i xng ca th Baứi Tỡm im un ca th cỏc hm s sau: a) y = x - x + x + b) y = x - x - x + c) y = x - x + x4 - 2x2 + e) y = x - 12 x + 48 x + 10 f) y = x - x + x - Baứi Tỡm m, n th ca hm s sau cú im un c ch ra: d) y = x3 + (m - 1) x + (m + 3) x - ; I(1; 3) 3 ổ2 d) y = x - mx + nx - ; I ỗ ; -3 ữ ố3 ứ a) y = x - x + 3mx + 3m + ; I(1; 2) b) y = c) y = mx + nx + ; I(1; 4) x3 e) y = - + 3mx - ; I(1; 0) f) y = mx + 3mx + ; I(1; 2) m Baứi Tỡm m th ca cỏc hm s sau cú im un: x5 4 x + mx - - x + (4m + 3) x3 + x - b) y = x2 + Baứi Chng minh th ca cỏc hm s sau cú im un thng hng: a) y = a) y = d) y = g) y = 2x +1 x2 + x + 2x +1 x2 + x - 3x b) y = e) y = x +1 x2 + x x2 + x2 + 3x h) y = x2 - 3x + x2 + Baứi Tỡm m, n th ca cỏc hm s: c) y = f) y = i) y = x - 3x x2 + x2 + x + x2 - x +1 x3 x2 - x + a) y = x - x - x + mx + m - cú hai im un thng hng vi im A(1; 2) x3 - x + mx + cú im un trờn ng thng y = x + 3 c) y = - x + mx + n cú im un trờn Ox b) y = - Trang 15 Kho sỏt hm s Trn S Tựng V NG TIM CN CA TH nh ngha: ã ng thng x = x0 gl ng tim cn ng ca th hm s y = f ( x ) nu ớt nht mt cỏc iu kin sau c tho món: lim + f ( x ) = +Ơ ; lim + f ( x ) = -Ơ ; lim - f ( x ) = +Ơ ; lim - f ( x ) = -Ơ xđ x0 xđ x0 xđ x0 xđ x0 ã ng thng y = y0 gl ng tim cn ngang ca th hm s y = f ( x ) nu ớt nht mt cỏc iu kin sau c tho món: lim f ( x ) = y0 ; lim f ( x ) = y0 x đ+Ơ x đ-Ơ ã ng thng y = ax + b, a gl ng tim cn xiờn ca th hm s y = f ( x ) nu ớt nht mt cỏc iu kin sau c tho món: lim x đ+Ơ [ f ( x ) - (ax + b)] = ; lim x đ-Ơ [ f ( x ) - (ax + b)] = Chỳ ý: a) Nu y = f ( x ) = P( x ) l hm s phõn thc hu t Q( x ) ã Nu Q(x) = cú nghim x0 thỡ th cú tim cn ng x = x0 ã Nu bc(P(x)) Ê bc(Q(x)) thỡ th cú tim cn ngang ã Nu bc(P(x)) = bc(Q(x)) + thỡ th cú tim cn xiờn b) xỏc nh cỏc h s a, b phng trỡnh ca tim cn xiờn, ta cú th ỏp dng cỏc cụng thc sau: f ( x) ; a = lim b = lim [ f ( x ) - ax ] x đ+Ơ x x đ+Ơ hoc a = lim x đ-Ơ f ( x) ; x b = lim x đ-Ơ [ f ( x ) - ax ] Baứi Tỡm cỏc tim cn ca th cỏc hm s sau: 2x - 10 x + b) y = x -1 1- 2x x - 4x + ( x - 2)2 d) y = e) y = x +1 1- x Baứi Tỡm cỏc tim cn ca th cỏc hm s sau: a) y = a) y = d) y = x x2 - x + x2 + 3x + b) y = 2+x - x2 x3 + x + e) y = x2 + x + x2 + Baứi Tỡm cỏc tim cn ca th cỏc hm s sau: 4x + a) y = x - x b) y = x2 - Trang 16 2x + 2- x 7x2 + x + f) y = - 3x c) y = c) y = f) y = c) y = x2 + x + x2 - x4 - x + x3 - 1 x2 - 4x + Trn S Tựng Kho sỏt hm s x -1 e) y = x - x x +1 Baứi Tỡm cỏc tim cn ca th cỏc hm s sau: d) y = x f) y = x - 3x + x-2 e x - e- x c) y = ln( x - x + 6) x 2 -1 Baứi Tỡm m th ca cỏc hm s sau cú ỳng hai tim cn ng: a) y = a) y = d) y = 2x + b) y = ln x + 2(2 m + 3) x + m - x -3 b) y = + x2 x + 2(m + 1) x + x -1 c) y = x +3 x + x +m-2 e) y = f) y = x + mx + m - x + 2(m + 2) x + m + x + 2(m - 1) x + m2 - Baứi Tỡm m th ca cỏc hm s sau cú tim cn xiờn: x + (3m + 2) x + m - mx + (2 m + 1) x + m + a) y = b) y = x+5 x+2 Baứi Tớnh din tớch ca tam giỏc to bi tim cn xiờn ca th cỏc hm s sau chn trờn hai trc to : 3x2 + x + -3 x + x - x2 + x - b) y = c) y = a) y = x -1 x+2 x -3 Baứi Tỡm m tim cn xiờn ca th cỏc hm s sau to vi cỏc trc to mt tam giỏc cú din tớch S ó ch ra: x + (2m - 1) x - m + x + mx - y a) y = ;S=8 b) = ;S=8 x +1 x -1 x + 2(2 m + 1) x + 4m - x + mx - c) y = ; S = 16 d) y = ;S=4 x +1 x -1 Baứi Chng minh rng tớch cỏc khong cỏch t mt im bt kỡ trờn th ca cỏc hm s n hai tim cn bng mt hng s: a) y = x2 - x + x -1 b) y = x2 + 5x - x +3 Trang 17 c) y = x2 + x - x -3 Kho sỏt hm s Trn S Tựng VI KHO ST S BIN THIấN V V TH CA HM S Cỏc bc kho sỏt s bin thiờn v v th ca hm s ã Tỡm xỏc nh ca hm s ã Xột s bin thiờn ca hm s: + Tớnh y + Tỡm cỏc im ti ú o hm y bng hoc khụng xỏc nh + Tỡm cỏc gii hn ti vụ cc, gii hn vụ cc v tỡm tim cn (nu cú) + Lp bng bin thiờn ghi rừ du ca o hm, chiu bin thiờn, cc tr ca hm s ã V th ca hm s: + Tỡm im un ca th (i vi hm s bc ba v hm s trựng phng) Tớnh y Tỡm cỏc im ti ú y = v xột du y + V cỏc ng tim cn (nu cú) ca th + Xỏc nh mt s im c bit ca th nh giao im ca th vi cỏc trc to (trong trng hp th khụng ct cỏc trc to hoc vic tỡm to giao im phc thỡ cú th b qua) Cú th tỡm thờm mt s im thuc th cú th v chớnh xỏc hn + Nhn xột v th: Ch trc i xng, tõm i xng (nu cú) ca th Hm s bc ba y = ax + bx + cx + d (a 0) : ã Tp xỏc nh D = R ã th luụn cú mt im un v nhn im un lm tõm i xng ã Cỏc dng th: a>0 y = cú nghim phõn bit y = b2 3ac > a0 a 0 y x ax + b (c 0, ad - bc 0) : cx + d ỡ dỹ ã Tp xỏc nh D = R \ ớ- ý ợ cỵ Hm s nht bin y = d a v mt tim cn ngang l y = Giao im ca c c hai tim cn l tõm i xng ca th hm s ã Cỏc dng th: ã th cú mt tim cn ng l x = - y y 0 x ad bc > x ad bc < ax + bx + c (a.a ' 0, tửỷ khoõng chia heỏt cho maóu) : a' x + b' ỡ b'ỹ ã Tp xỏc nh D = R \ ớ- ý ợ a'ỵ Hm s hu t y = ã th cú mt tim cn ng l x = - b' v mt tim cn xiờn Giao im ca hai tim a' cn l tõm i xng ca th hm s ã Cỏc dng th: a.a > Trang 19 a.a < Kho sỏt hm s Trn S Tựng y = cú nghim phõn bit y y = vụ nghim y x x Baứi Kho sỏt s bin thiờn v v th ca cỏc hm s: a) y = x - x - x + b) y = x + x + x + x3 - x2 + 3 Baứi Kho sỏt s bin thiờn v v th ca cỏc hm s: d) y = ( x - 1)2 (4 - x ) a) y = x - x - e) y = b) y = x - x + d) y = ( x - 1)2 ( x + 1)2 e) y = - x + x + Baứi Kho sỏt s bin thiờn v v th ca cỏc hm s: x +1 2x +1 a) y = b) y = x -1 x +2 1- 2x 3x - d) y = e) y = 1+ 2x x -3 Baứi Kho sỏt s bin thiờn v v th ca cỏc hm s: x2 + x + a) y = x +1 x -1 Baứi V th ca cỏc hm s: d) y = - x + + a) y = x - x + d) y = x +1 x -1 x2 + x + b) y = x -1 e) y = x2 1- x b) y = - x + x - e) y = x2 - x + x -1 Trang 20 c) y = - x + x - f) y = - x3 - x - x + x4 c) y = - 3x2 + 2 f) y = -2 x + x + 3- x x -4 x -2 f) y = 2x +1 c) y = x2 + x - c) y = x +1 f) y = x2 - 2x x +1 c) y = x - x - f) y = x2 + 3x + x+2