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AI & ML · Multivariable Calculus

Jacobians & Hessians

80 problems in this skill, each with a solver-verified answer key. Answers are checked the moment you submit — with a hint if you want one and a worked solution afterwards.

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

  1. Warm-up

    A scalar function is f(x,y)=−x2−2y2+3x+yf(x,y) = -x^{2} - 2y^{2} + 3x + y. Find the determinant of its Hessian matrix H=(fxxfxyfyxfyy)H=\begin{pmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{pmatrix}.

  2. Warm-up

    A scalar function is f(x,y)=−3x2+2xy−4y2−2x+2yf(x,y) = -3x^{2} + 2xy - 4y^{2} - 2x + 2y. Find the determinant of its Hessian matrix H=(fxxfxyfyxfyy)H=\begin{pmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{pmatrix}.

  3. Warm-up

    A scalar function is f(x,y)=−x2+2xy−y2+2x+2yf(x,y) = -x^{2} + 2xy - y^{2} + 2x + 2y. Find the determinant of its Hessian matrix H=(fxxfxyfyxfyy)H=\begin{pmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{pmatrix}.

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