Question

If $D=4a^2-12b=4(a^2-3b)=0$ and let $x_1$ and $x_2$ be the roots of $f^{\prime }\left(x\right)=0$ such that $f\left(x_1\right)\ne 0$, then

• A. $f\left(x\right)$ has all real roots
• B. $f\left(x\right)$ has one real and two non-real roots
• C. $f\left(x\right)$ has repeated roots
• D. none of the above

Answer 1

The correct option is B $f\left(x\right)$ has one real and two non-real roots

$D=0$
$\Rightarrow x_1=x_2$
Hence, $f^{\prime }\left(x\right)$ is always non-negative. Hence, the function is monotonic increasing .
As $f\left(x_1\right)\ne 0$, roots are not repeated.
Hence, there is one real root and two imaginary roots.