Exercise · ZW-LOCAL-AND-GLOBAL-MINIMA-OF-CONVEX-FUNCTIONS-009 · ★★☆ Intermediate
Let \(f(x,y)=x^2+y^2-2x\) over \(\mathbb{R}^2\). What is the global minimum?
Choose one option.
Show answer
- D. -1 at (1,0)
Explanation
Written as \((x-1)^2+y^2-1\), the minimum is -1 at (1,0). The option with -2 forgets the constant term when completing the square.
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-002 Is the point x=3 a global minimum of f(x)=(x-3)^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?
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