Question
The function $$f\left( x \right) = {x^3} + \lambda {x^2} + 5x + \sin \,2x$$ will be an invertible function if $$\lambda $$ belongs to :
A.
$$\left( { - \infty ,\, - 3} \right)$$
B.
$$\left( { - 3,\,3} \right)$$
C.
$$\left( {3,\, + \infty } \right)$$
D.
none of these
Answer :
$$\left( { - 3,\,3} \right)$$
Solution :
$$\eqalign{
& f'\left( x \right) = 3{x^2} + 2\lambda x + 5 + 2\cos \,2x \leqslant 3{x^2} + 2\lambda x + 7\,\,\,\,\left( {\because \max \,\cos \,2x = 1} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \geqslant 3{x^2} + 2\lambda x + 3\,\,\,\left( {\because \min \,\cos \,2x = - 1} \right) \cr
& {\text{But }}3{x^2} + 2\lambda x + 7 < 0{\text{ is not true for all}}\;x\, \in \,R \cr
& \therefore \,f'\left( x \right) \geqslant 3{x^2} + 2\lambda x + 3 > 0{\text{ for all }}x{\text{ if }}D < 0,{\text{ i}}{\text{.e}}{\text{., }}4{\lambda ^2} - 4.3.3 < 0 \cr
& {\text{or, }}{\lambda ^2} - 9 < 0\,\,\,\, \Rightarrow - 3 < \lambda < 3 \cr} $$
$$\therefore $$ if $$ - 3 < \lambda < 3,\,f\left( x \right)$$ is strictly $$m.i.$$ and so $$f\left( x \right)$$ is invertible.