Let $f(x)=100x^3-300x^2+200x$. For how many real numbers $a$ does the graph of $y=f(x-a)$ pass through the point $(1,25)$?
$$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } \text{more than } 4$$
Difficulty: 45/100 — Unprecedentedly hard for a problem 9; quite involved with cubics and root finding. Quite hard to tackle for beginners and extremely misplaced.
Core Concepts: functions, transformations, roots
Challenges: Recognizing that the condition is equivalent to asking how many $x$ satisfy $f(x)=25$. Interpreting the graph of a cubic. Solving for number of roots without directly finding them.
Techniques: Calculate maxima and minima to count intersections.
Error-prone Steps: Attempting exact roots unnecessarily, applying calculus, lazily assuming there are three intersection points because it is a cubic.
Ideal Time:
Experienced: ≤ 2-3 min
Intermediate: ≤ 4 min
Beginner: ≤ 10 min