Theory of Equations 5 Question 4

4. The real number $k$ for which the equation, $2 x^{3}+3 x+k=0$ has two distinct real roots in $[0,1]$

(2013 Main)

(a) lies between 1 and 2

(b) lies between 2 and 3

(c) lies between -1 and 0

(d) does not exist

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Answer:

Correct Answer: 4. (d)

Solution:

  1. Let $f(x)=2 x^{3}+3 x+k$

On differentiating w.r.t. $x$, we get

$ f^{\prime}(x)=6 x^{2}+3>0, \forall x \in R $

$\Rightarrow f(x)$ is strictly increasing function.

$\Rightarrow f(x)=0$ has only one real root, so two roots are not possible.



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