Let f(.) be a twice differentiable function from $\R^2\rightarrow\R$ . If p, x 0 ∈ $\R^2$ 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 x 0 and x 0 + p. If x 0 is a strict local minimum of f(x), then which one of the following statements is TRUE?

Question

Let f(.) be a twice differentiable function from $\R^2\rightarrow\R$ . If p, x 0 ∈  $\R^2$  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 x 0 and x 0 + p. If x 0 is a strict local minimum of f(x), then which one of the following statements is TRUE?

cdquestions Admin · Accepted Answer

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

Answer

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

Answer

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

Answer

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

Answer

$\triangledown f(x_0)^Tp \gt0\ \text{and}\ p^T\triangledown^2f(\psi)p\lt0$

The Correct Option is B

Solution and Explanation

Top Questions on Total derivative

Questions Asked in GATE Metallurgy exam