Exercise · ZW-LOCAL-AND-GLOBAL-MINIMA-OF-CONVEX-FUNCTIONS-002 · ★☆☆ Basic
Is the point \(x=3\) a global minimum of \(f(x)=(x-3)^2\) over \(\mathbb{R}\)?
Choose one option.
Show answer
- C. Yes, because f(3)=0 and f(x)≥0 for all x
Explanation
The function is convex and its value at \(x=3\) is 0, which is never undercut. The option saying the gradient is nonzero confuses the gradient at the point with the optimality condition.
Review the concept: Local and global minima of convex functions
A local minimum of a function is a point where no sufficiently small neighborhood contains smaller values, while a global minimum is a point where the function attains the lowest value over its entire domain.
\[f(x^*) \le f(x) \quad \forall x \in \mathrm{dom}\, f\]
More practice problems: local and global minima of convex functions
- ZW-LOCAL-AND-GLOBAL-MINIMA-OF-CONVEX-FUNCTIONS-001 What is the minimum value of f(x)=x^2 over ℝ?
- ZW-LOCAL-AND-GLOBAL-MINIMA-OF-CONVEX-FUNCTIONS-003 For f(x)=x^4, what is the global minimum over ℝ?
- ZW-LOCAL-AND-GLOBAL-MINIMA-OF-CONVEX-FUNCTIONS-004 Which set describes the minimizers of f(x)=|x| over ℝ?
- ZW-LOCAL-AND-GLOBAL-MINIMA-OF-CONVEX-FUNCTIONS-005 If a convex function has a local minimum at x_0, what can be said?
- ZW-LOCAL-AND-GLOBAL-MINIMA-OF-CONVEX-FUNCTIONS-006 Let f(x)=(x-2)^2+1 on [0,5]. What is the global minimum and where is it attained?
- ZW-LOCAL-AND-GLOBAL-MINIMA-OF-CONVEX-FUNCTIONS-007 Let f(x)=x^2-4x over ℝ. What is the global minimum?
- ZW-LOCAL-AND-GLOBAL-MINIMA-OF-CONVEX-FUNCTIONS-008 Let f(x)=e^x on [-1,2]. What is the global minimum?
- ZW-LOCAL-AND-GLOBAL-MINIMA-OF-CONVEX-FUNCTIONS-009 Let f(x,y)=x^2+y^2-2x over ℝ^2. What is the global minimum?
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