Tìm tất cả các giá trị thực của m để hàm số [y = {2^{{x^3} - {x^2} + mx + 1}} ] đồng biến trên (1;2) A. [m > - 8. ] B. [m ge - 1. ] C. [m le - 8. ] D. [m < - 1. ]

Ta có: [y = {2^{{x^3} - {x^2} + mx + 1}} Rightarrow y' = left( {3{x^2} - 2x + m} right){2^{{x^3} - {x^2} + mx + 1}} ] ⇒ Hàm số đã cho đồng biến trên [ left( {1; , ,2} right) Leftrightarrow y' ge 0 , , forall x in left( {1; , ,2} right) ] [ Leftrightarrow (3{x^2} - 2x + m){2^{{x^3} - x2 + mx + 1}} ge 0 forall x in (1;2) ] [ Leftrightarrow 3{x^2} - 2x + m ge 0 forall x in (1;2) ] ( Leftrightarrow { left[ { begin{array}{*{20}{c}}{ Delta prime le 0} { left { { begin{array}{*{20}{c}}{ Delta prime ge 0} { left[ { begin{array}{*{20}{c}}{{x_1} < {x_2} le 1} {2 le {x_1} < x2} end{array}} right.} end{array}} right.} end{array}} right._{}} Leftrightarrow left[ { begin{array}{*{20}{c}}{ Delta prime le 0} { left { { begin{array}{*{20}{c}}{ Delta prime ge 0} { left[ { begin{array}{*{20}{c}}{ left { { begin{array}{*{20}{c}}{{x_1} + {x_2} < 2} {({x_1} - 1)({x_2} - 1) ge 0} end{array}} right.} { left { { begin{array}{*{20}{c}}{{x_1} + {x_2} > 4} {({x_1} - 1)({x_2} - 1) ge 0} end{array}} right.} end{array}} right.} end{array}} right.} end{array}} right. )</> ( Leftrightarrow left[ { begin{array}{*{20}{c}}{ Delta prime le 0} { left { { begin{array}{*{20}{c}}{ Delta prime ge 0} { left[ { begin{array}{*{20}{c}}{ left { { begin{array}{*{20}{c}}{{x_1} + {x_2} < 2} {{x_1}{x_2} - ({x_1} + {x_2}) + 1 ge 0} end{array}} right.} { left { { begin{array}{*{20}{c}}{{x_1} + {x_2} > 4} {{x_1}{x_2} - ({x_1} + {x_2}) + 1 ge 0} end{array}} right.} end{array}} right.} end{array}} right.} end{array}} right. )</> ( Leftrightarrow left[ { begin{array}{*{20}{c}}{1 - 3m le 0} { left { { begin{array}{*{20}{c}}{1 - 3m ge 0} { left[ { begin{array}{*{20}{c}}{ frac{2}{3} < 2} { frac{m}{3} - frac{2}{3} + 1 ge 0} end{array}} right.} { left[ { begin{array}{*{20}{c}}{ frac{x}{3} > 4(ktm)} { frac{m}{3} - frac{4}{3} + 4 ge 0} end{array}} right.} end{array}} right.} end{array}} right. )</> ( Rightarrow left[ { begin{array}{*{20}{c}}{m ge frac{1}{3}} { left { { begin{array}{*{20}{c}}{m le frac{1}{3}} { frac{m}{3} ge - frac{1}{3}} end{array}} right.} end{array}} right. Leftrightarrow left[ { begin{array}{*{20}{c}}{m ge frac{1}{3}} { left { { begin{array}{*{20}{c}}{m le frac{1}{3}} {m ge - 1} end{array}} right.} end{array}} right. Leftrightarrow left[ { begin{array}{*{20}{c}}{m ge frac{1}{3}} { - 1 le m le frac{1}{3}} end{array} Leftrightarrow m ge - 1.} right. ) Đáp án cần chọn là: B