Cho hàm số \[y = f\left( x \right) = \left( {1 - 2{x^2}} \right)\sqrt {1 + 2{x^2}} \]. Ta xét hai mệnh đề sau:

     (I) \[f'\left( x \right) = \frac{{ - 2x\left( {1 + 6{x^2}} \right)}}{{\sqrt {1 + 2{x^2}} }}\]                            (II) \[f\left( x \right).f'\left( x \right) = 2x\left( {12{x^4} - 4{x^2} - 1} \right)\]

     Mệnh đề nào đúng?

Hướng dẫn giải:

Đáp án D

Ta có

\[\begin{array}{l}f'\left( x \right) = {\left( {1 - 2{x^2}} \right)^\prime }\sqrt {1 + 2{x^2}} + \left( {1 - 2{x^2}} \right){\left( {\sqrt {1 + 2{x^2}} } \right)^\prime } = - 4x\sqrt {1 + 2{x^2}} + \left( {1 - 2{x^2}} \right)\frac{{2x}}{{\sqrt {1 + 2{x^2}} }}\\\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{ - 4x\left( {1 + 2{x^2}} \right) + \left( {1 - 2{x^2}} \right).2x}}{{\sqrt {1 + 2{x^2}} }} = \frac{{ - 2x - 12{x^3}}}{{\sqrt {1 + 2{x^2}} }} = \frac{{ - 2x\left( {1 + 6{x^2}} \right)}}{{\sqrt {1 + 2{x^2}} }}\end{array}\]

Suy ra

 \[\begin{array}{l}f\left( x \right).f'\left( x \right) = \left( {1 - 2{x^2}} \right)\sqrt {1 + 2{x^2}} .\frac{{ - 2x\left( {1 + 6{x^2}} \right)}}{{\sqrt {1 + 2{x^2}} }} = - 2x\left( {1 - 2{x^2}} \right)\left( {1 + 6{x^2}} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - 2x\left( { - 12{x^4} + 4{x^2} + 1} \right) = 2x\left( {12{x^4} - 4{x^2} - 1} \right)\end{array}\]