1
GATE ME 2024
+1
-0.33

Let $f(.)$ be a twice differentiable function from $\mathbb{R}^{2} \rightarrow \mathbb{R}$. If $P, \mathbf{x}_{0} \in \mathbb{R}^{2}$ where $\vert \vert P\vert \vert$ is sufficiently small (here $\vert \vert . \vert \vert$ is the Euclidean norm or distance function), then $f (\mathbf{x}_{0} + p) = f(\mathbf{x}_{0}) + \nabla f(\mathbf{x}_{0})^{T}p + \dfrac{1}{2} p^{T} \nabla^{2}f(\psi)p$ where $\psi \in \mathbb{R}^{2}$ is a point on the line segment joining $\mathbf{x}_{0}$ and $\mathbf{x}_{0} + p$. If $\mathbf{x}_{0}$ is a strict local minimum of $f (\mathbf{x})$, then which one of the following statements is TRUE?

A

$\nabla f(x_{0})^{T}p > 0\ \ and\ \ p^{T} \nabla^{2} f( \psi)p = 0$

B

$\nabla f(x_{0})^{T}p = 0\ and\ p^{T} \nabla^{2} f( \psi)p > 0$

C

$\nabla f(x_{0})^{T}p = 0\ and\ p^{T} \nabla^{2} f( \psi)p = 0$

D

$\nabla f(x_{0})^{T}p = 0\ and\ p^{T} \nabla^{2} f( \psi)p < 0$

2
GATE ME 2020 Set 1
+1
-0.33
Define [x] as the greatest integer less than or equal to x, for each x ϵ (-∞, ∞). If y = [x], then area under y for x ϵ [1,4] is
A
1
B
3
C
4
D
6
3
GATE ME 2017 Set 1
+1
-0.3
The value of $$\mathop {\lim }\limits_{x \to 0} \left( {{{{x^3} - \sin \left( x \right)} \over x}} \right)$$ is
A
$$0$$
B
$$3$$
C
$$1$$
D
$$-1$$
4
GATE ME 2016 Set 2
+1
-0.3
The values of $$x$$ for which the function $$f\left( x \right) = {{{x^2} - 3x - 4} \over {{x^2} + 3x - 4}}$$ is NOT continuous are
A
$$4$$ and $$-1$$
B
$$4$$ and $$1$$
C
$$-4$$ and $$1$$
D
$$-4$$ and $$-1$$
GATE ME Subjects
EXAM MAP
Medical
NEET