MMiDS 3.2: Self-Assessment Quiz

What does it mean for a function \( f \) to be continuously differentiable at \( x_0 \)?

• a) \( f \) is continuous at \( x_0 \).
• b) All partial derivatives of \( f \) exist at \( x_0 \).
• c) All partial derivatives of \( f \) exist and are continuous in an open ball around \( x_0 \).
• d) The gradient of \( f \) is zero at \( x_0 \).

What is the gradient of a function \( f : D \to \mathbb{R} \), where \( D \subseteq \mathbb{R}^d \), at a point \( x_0 \in D \)?

• a) The rate of change of \( f \) with respect to \( x \) at \( x_0 \)
• b) The vector of all second partial derivatives of \( f \) at \( x_0 \)
• c) The vector of all first partial derivatives of \( f \) at \( x_0 \)
• d) The matrix of all second partial derivatives of \( f \) at \( x_0 \)

For a twice continuously differentiable function \( f \) at \( x_0 \), what is the Hessian matrix?

• a) The matrix of all first partial derivatives of \( f \) at \( x_0 \)
• b) The vector of all first partial derivatives of \( f \) at \( x_0 \)
• c) The matrix of all second partial derivatives of \( f \) at \( x_0 \)
• d) The vector of all second partial derivatives of \( f \) at \( x_0 \)

Consider the function \( f(x_1, x_2) = 2x_1^2 + 3x_2^2 - 2x_1x_2 \). What is the value of the partial derivative \( \frac{\partial f}{\partial x_1} \) at the point \( (1, 2) \)?

• a) 0
• b) 2
• c) 4
• d) 6

What is the gradient of the function \( f(x_1, x_2) = x_1^2 - 3x_1x_2 - x_2^3 \) at the point \( (1, -1) \)?

• a) \( (5, -6) \)
• b) \( (5, 6) \)
• c) \( (-5, -6) \)
• d) \( (-5, 6) \)

Which of the following statements is true about the Hessian matrix of a twice continuously differentiable function?

• a) It is always a diagonal matrix.
• b) It is always a symmetric matrix.
• c) It is always an invertible matrix.
• d) It is always a positive definite matrix.

If \( f(x, y) = e^{x^2 + y} \), what is \( \frac{\partial^2 f}{\partial x \partial y} \)?

• a) \( 2xe^{x^2 + y} \)
• b) \( 2x \)
• c) \( e^{x^2 + y} \)
• d) 0

For the function \( f(x_1, x_2) = x_1^3 + 2x_1x_2^2 \), calculate the Hessian matrix \( \mathbf{H}_f(1, 1) \).

• a) \( \begin{pmatrix} 6 & 4 \\ 4 & 4 \end{pmatrix} \)
• b) \( \begin{pmatrix} 3 & 2 \\ 2 & 2 \end{pmatrix} \)
• c) \( \begin{pmatrix} 6 & 2 \\ 2 & 4 \end{pmatrix} \)
• d) \( \begin{pmatrix} 3 & 4 \\ 4 & 2 \end{pmatrix} \)

Let \( f(x, y, z) = x^2 + y^2 - z^2 \). What is the Hessian matrix of \( f \)?

• a) \( \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -2 \end{pmatrix} \)
• b) \( \begin{pmatrix} 2x & 2y & -2z \end{pmatrix} \)
• c) \( \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \)
• d) \( \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \)

The multivariable version of the Mean Value Theorem states that for a continuously differentiable function \( f : D \to \mathbb{R} \), where \( D \subseteq \mathbb{R}^d \), and \( x_0, x \in D \) with \( \mathbf{x} \in B_\delta(\mathbf{x}_0) \) for some \( \delta > 0 \), there exists \( \xi \in (0, 1) \) such that:

• a) \( f(\mathbf{x}) = f(\mathbf{x}_0) + \nabla f(\mathbf{x}_0)^T(\mathbf{x}_0 + \xi(\mathbf{x} - \mathbf{x}_0)) \)
• b) \( f(\mathbf{x}) = f(\mathbf{x}_0) + \nabla f(\mathbf{x}_0 + \xi(\mathbf{x} - \mathbf{x}_0))^T(\mathbf{x} - \mathbf{x}_0) \)
• c) \( f(\mathbf{x}) = f(\mathbf{x}_0) + \nabla f(\mathbf{x} + \xi(\mathbf{x} - \mathbf{x}_0))^T(\mathbf{x} - \mathbf{x}_0) \)
• d) \( f(\mathbf{x}) = f(\mathbf{x}_0 + \xi(\mathbf{x} - \mathbf{x}_0)) + \nabla f(\mathbf{x}_0)^T(\mathbf{x} - \mathbf{x}_0) \)

What is the gradient of the affine function \( f(\mathbf{x}) = \mathbf{q}^T\mathbf{x} + r \), where \( \mathbf{x} = (x_1, \ldots, x_d) \) and \( \mathbf{q} = (q_1, \ldots, q_d) \in \mathbb{R}^d \)?

• a) \( \nabla f(\mathbf{x}) = \mathbf{q}^T \)
• b) \( \nabla f(\mathbf{x}) = \mathbf{x} \)
• c) \( \nabla f(\mathbf{x}) = \mathbf{q} \)
• d) \( \nabla f(\mathbf{x}) = r \)

What is the Hessian matrix of the quadratic function \( f(\mathbf{x}) = \frac{1}{2}\mathbf{x}^T P \mathbf{x} + \mathbf{q}^T \mathbf{x} + r \), where \( P \in \mathbb{R}^{d \times d} \) and \( \mathbf{q} \in \mathbb{R}^d \)?

• a) \( \mathbf{H}_f(\mathbf{x}) = P \)
• b) \( \mathbf{H}_f(\mathbf{x}) = P^T \)
• c) \( \mathbf{H}_f(\mathbf{x}) = \frac{1}{2}[P + P^T] \)
• d) \( \mathbf{H}_f(\mathbf{x}) = [P + P^T] \)

Let \( f(x, y) = x^2y + 3y^2 \). If \( \mathbf{g}(t) = (t, t^2) \), what is \( (f \circ \mathbf{g})'(1/2) \)?

• a) 7
• b) 8
• c) 9
• d) 10