Question 1 · Optimization: Local and Global Extrema · 2025
MSQ
Correct Answer:
["B","C","D"]
Step-by-Step Solution
Insight: Strict convexity means the function curves strictly upwards, preventing multiple flat spots or multiple dips, but not guaranteeing a minimum exists.
Exam route: Use the definition of strict convexity. (A) is false because is strictly convex but has no minimum. (B) is true because the derivative is strictly increasing. (C) and (D) are true because strict convexity forbids multiple minima.
Learning route:
- Define strict convexity: for and .
- Tempting wrong path: "Convex means bowl-shaped, so it must have a bottom." This leads to selecting (A). This breaks at the counterexample , which is strictly convex but strictly increasing, so it never flattens out. Generalization: Strict convexity guarantees the shape, but not the existence of an extremum unless the function is coercive.
- Check (A): False, as shown by .
- Check (B): If is strictly convex and differentiable at , then . Thus, can be zero at most once. There do not exist with . (B) is true.
- Check (C): Suppose two global minima exist with . Then , contradicting is the global minimum. (C) is true.
- Check (D): For convex functions, any local minimum is a global minimum. Since there is at most one global minimum, there is at most one local minimum. (D) is true.
Verification: For , (B holds), no global min (C holds), no local min (D holds), no local min (A fails).