Question

Interval in which $‘a^{\prime }$ lies so that $x^3–3x+a=0$ has three real and distinct roots.

• A. (-1, 1)
• B. (-2, 2)
• C. (-1 ,2)
• D. (-2, 1)

Answer 1

The correct option is B

(-2, 2)

In this question, we want to find the nature of the roots of cubic equation. We can do such problems using the concepts from theory of equations or by taking derivatives. When we differentiate this cubic polynomial, we get a quadratic equation. Now, if the quadratic equation has no real roots, we can say that the cubic equation will have only one root. If

Let $f\left(x\right)=x^3–3x+a$

$f^{\prime }\left(x\right)=3x^2-3=3(x-1)(x+1)$

Clearly $x=-1$ is the point of maximum and $x=1$ is the point of minimum.

Now, $f\left(1\right)=a-2,$

$f(-1)=a+2$

The roots of $f\left(x\right)=0$ would be real and distinct if $f\left(1\right)f(-1)<0$

$=(a-2)(a+2)<0$

$=-2<a<2$

Thus given equation would have real and distinct roots if a $\in (-2,2)$