Question

lf $f\left(x\right)=0$ has a repeated root $\alpha$, then another equation having $\alpha$ as root, is

• A. $f\left(2x\right)=0$
• B. $f\left(3x\right)=0$
• C. $f^{{}^{\prime }}\left(x\right)=0$
• D. $f^{{}^{\prime \prime }}\left(x\right)=0$

Answer 1

The correct option is C $f^{{}^{\prime }}\left(x\right)=0$

Repeated roots mean that the curve will touch the $x$-axis.
The curve will have a convex or concave part at the repeated root $\alpha$ or a point of inflection.
In any case $f^{\prime }\left(x\right)$ will always be zero at the point of repeated root $\alpha$.