Mathematics

The key is recognizing that $x=1$ is a repeated root of the derivative, which tells you something special about the function's behavior there. Even though $f'(1)=0$, you need to look at higher derivatives or sketch the sign of $f'(x)$ around $x=1$ to see whether the function actually changes direction—or just "flattens out" while continuing in the same direction. A repeated critical point often signals an inflection point rather than a turning point.

The key is recognizing that $x=1$ is a repeated root of the derivative, which tells you something special about the function's behavior there. Even though $f'(1)=0$, you need to look at higher derivatives or sketch the sign of $f'(x)$ around $x=1$ to see whether the function actually changes direction—or just "flattens out" while continuing in the same direction. A repeated critical point often signals an inflection point rather than a turning point.