Question

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?

• A. $\triangledown f(x_0)^Tp \gt0\ \text{and}\ p^T\triangledown^2f(\psi)p=0$
• B. $\triangledown f(x_0)^Tp =0\ \text{and}\ p^T\triangledown^2f(\psi)p\gt0$
• C. $\triangledown f(x_0)^Tp =0\ \text{and}\ p^T\triangledown^2f(\psi)p=0$
• D. $\triangledown f(x_0)^Tp \gt0\ \text{and}\ p^T\triangledown^2f(\psi)p\lt0$

Answer 1

Answer: (B)

$\triangledown f(x_0)^Tp =0\ \text{and}\ p^T\triangledown^2f(\psi)p\gt0$

Answer 2

The correct option is (B): $\triangledown f(x_0)^Tp =0\ \text{and}\ p^T\triangledown^2f(\psi)p\gt0$