MMiDS 3.2: Self-Assessment Quiz

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

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

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

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 \( (1, 2) \)?

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

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

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

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

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

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:

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 \( \mathbf{q} = (q_1, \ldots, q_d) \in \mathbb{R}^d \)?

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 \( \mathbf{q} \in \mathbb{R}^d \)?

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