Exercise · ZW-CONVEXITY-AND-THE-HESSIAN-007 · ★★☆ Intermediate
For which values of \(a\) is \(f(x,y)=x^2+axy+y^2\) convex on \(\mathbb{R}^2\)?
Choose one option.
Show answer
- B. \(|a|\leq 2\)
Explanation
The Hessian is \(\begin{pmatrix}2&a\\a&2\end{pmatrix}\), with eigenvalues \(2\pm a\), which are nonnegative exactly when \(|a|\leq 2\). The condition \(a\geq 0\) ignores the negative case.
Review the concept: Convexity and the Hessian
Criterion characterizing convexity of a twice differentiable function through positive semidefiniteness of its Hessian matrix \(\nabla^2 f(x)\) at every point of the domain.
\[f \text{ convexa} \iff \nabla^2 f(x) \succeq 0 \ \forall x \in \operatorname{dom} f\]
More practice problems: convexity and the hessian
- ZW-CONVEXITY-AND-THE-HESSIAN-001 What condition on the Hessian guarantees that f is convex?
- ZW-CONVEXITY-AND-THE-HESSIAN-002 If ∇^2 f(x) is negative semidefinite everywhere, what kind of function is f?
- ZW-CONVEXITY-AND-THE-HESSIAN-003 For f(x)=x^2, what is its Hessian ∇^2 f(x)?
- ZW-CONVEXITY-AND-THE-HESSIAN-004 What does it mean for a symmetric matrix to be positive semidefinite?
- ZW-CONVEXITY-AND-THE-HESSIAN-005 What kind of set must the domain of f be to apply the Hessian criterion?
- ZW-CONVEXITY-AND-THE-HESSIAN-006 Compute the Hessian of f(x,y)=x^2+3xy and determine whether it is positive semidefinite.
- ZW-CONVEXITY-AND-THE-HESSIAN-008 Let f(x,y)=e^x+y^2. Is f convex on ℝ^2?
- ZW-CONVEXITY-AND-THE-HESSIAN-009 Let f(x,y)=x^3+y^2. What can be concluded about the convexity of f on \x≥ 0\?
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