List of practice Questions

The second derivative of a function f is computed using the fourth-order Central Divided Difference method with a step length h.
The CORRECT expression for the second derivative is

Let f(.) be a twice differentiable function from \(\R^2\rightarrow\R\). If p, x₀ ∈ \(\R^2\) where ||p|| is sufficiently small (here ||. || is the Euclidean norm or distance function), then \(f(x_0 + p) =f(x_0) + \triangledown f(x_0)^Tp+\frac{1}{2}p^T\triangledown^2f(\psi)p\) where \(\psi\isin\R^2\) is a point on the line segment joining x₀ and x₀ + p. If x₀ is a strict local minimum of f(x), then which one of the following statements is TRUE?

The family of surfaces given by \[ u = xy + f(x^2 - y^2), \text{where} \, f : \mathbb{R} \to \mathbb{R} \, \text{is a differentiable function, satisfies:} \]