Convexity

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   Let $f(x,y) = 5x^{2} + bxy + 4y^{2}$, so its Hessian is $H = \begin{pmatrix} 10 & b \\ b & 8 \end{pmatrix}$ with $\det(H) = 80 - b^{2}$. The boundary between $f$ being convex (positive semi-definite Hessian) and $f$ being a saddle (indefinite Hessian) occurs where $\det(H) = 0$. Find both values of $b$ at this boundary, separated by a comma.

2. Warm-up

   Let $f(x,y) = 7x^{2} + bxy + 9y^{2}$, so its Hessian is $H = \begin{pmatrix} 14 & b \\ b & 18 \end{pmatrix}$ with $\det(H) = 252 - b^{2}$. The boundary between $f$ being convex (positive semi-definite Hessian) and $f$ being a saddle (indefinite Hessian) occurs where $\det(H) = 0$. Find both values of $b$ at this boundary, separated by a comma.

3. Warm-up

   Let $f(x,y) = 3x^{2} + bxy + 2y^{2}$, so its Hessian is $H = \begin{pmatrix} 6 & b \\ b & 4 \end{pmatrix}$ with $\det(H) = 24 - b^{2}$. The boundary between $f$ being convex (positive semi-definite Hessian) and $f$ being a saddle (indefinite Hessian) occurs where $\det(H) = 0$. Find both values of $b$ at this boundary, separated by a comma.

Problems shown under their original terms: Generated (template, solver-computed).