Question

If $D=4(a^2-3b)>0$ and $f\left(x_1\right).f\left(x_2\right)<0$ where $x_1$, $x_2$ are the roots of $f^{\prime }\left(x\right)=0$, then

• A. $f\left(x\right)$ has all real and distinct roots
• B. $f\left(x\right)$ has three real roots but one of the roots would be repeated
• C. $f\left(x\right)$ would have just one real root
• D. none of the above

Answer 1

Answer: (A)

$f\left(x\right)$ has all real and distinct roots

$f^{\prime }\left(x\right)=0$ has real roots $x_1$ and $x_2$
Now, $x_1$ and $x_2$ would be points of maxima and minima (not in any order).
If $f\left(x_1\right).f\left(x_2\right)<0$, both $f\left(x_1\right)$ and $f\left(x_2\right)$ would have opposite signs. Hence, the maxima or the minima would lie on opposite side of the x-axis. Hence, one root will lie $x_1$ and $x_2$.
One root will lie between $(-\infty ,x_1)$ and other lie will lie in $(x_2,\infty )$
Hence, there will have three real roots.