Question

Find the value of a such that $x^3-\left|a\right|x^2+3x+4=0$ has only one real root.

Answer 1

For only one root it should be either strictly increasing or decreasing from $(-\infty ,\infty )$.for that we need $f^{\prime }\left(x\right)$;

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

we know that for a quadratic equation $a{x}^2+bx+c$ if $a>0,b^2-4ac<0$ it lies above $x-axis$.

using this condition for $f^{\prime }\left(x\right)>0$ as $3>0$

$(2|a|{)}^2-4\times 3\times 3<0$

$a^2-3^2=(a-3)(a+3)<0$

$aϵ(-3,3)$.