Question
The real number $$k$$ for which the equation, $$2{x^3} + 3x + k = 0$$ has two distinct real roots in [0, 1]
A.
lies between 1 and 2
B.
lies between 2 and 3
C.
lies between $$- 1$$ and 2
D.
does not exist.
Answer :
does not exist.
Solution :
$$\eqalign{
& f\left( x \right) = 2{x^3} + 3x + k \cr
& f'\left( x \right) = 6{x^2} + 3 > 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall \,x \in R\,\,\left( {\because \,\,{x^2} > 0} \right) \cr} $$
$$ \Rightarrow \,\,f\left( x \right)$$ is strictly increasing function
$$ \Rightarrow \,\,f\left( x \right) = 0$$ as only one real root, so two roots are not possible.