04 Gradients and Optimization Online Quiz Questions

True or false: At a point where $\nabla f\neq \mathbf{0}$, the gradient points in the direction of the steepest ascent of $f$.

For a differentiable function $f(x,y)$, what geometric role does $\nabla f$ play at a regular point of the level curve $f(x,y)=c$?

For $f(x,y)=x^2+xy$, find the directional derivative at $(1,2)$ in the direction of $\mathbf{v}=\langle 3,4\rangle$. Enter the exact value; the direction vector must first be normalized.

Select all statements that correctly describe the one-constraint Lagrange multiplier method for optimizing $f$ subject to $g=c$.

True or false: If $\nabla f(\mathbf{a})=\mathbf{0}$, then $\mathbf{a}$ must be a local maximum or local minimum.

Let $z=f(x,y)$, where $f(1,1)=3$, $f_x(1,1)=4$, and $f_y(1,1)=2$. Which equation is the tangent plane at $(1,1,3)$?

What is the standard name for the scalar $\lambda$ in the equation $\nabla f=\lambda\nabla g$?

Select all points at which $f(x,y)=xy$ attains its maximum subject to $x^2+y^2=1$.

At a critical point of a function of two variables, suppose $D=f_{xx}f_{yy}-(f_{xy})^2<0$. What does the second derivative test conclude?

Which unit vector gives the direction of $\mathbf{v}=\langle 3,4\rangle$ for use in a directional derivative?

Explain how to construct the tangent plane to the level surface $F(x,y,z)=c$ at a regular point $P=(x_0,y_0,z_0)$. State the necessary regularity condition and give the tangent-plane equation.

Let $f(x,y)=x^2+3y$. What is the directional derivative of $f$ at $(1,2)$ in the direction of $\mathbf{v}=\langle 3,4\rangle$?

A direction is represented by the vector $\mathbf{v}=\langle 3,4\rangle$. What is $\|\mathbf{v}\|$, the value needed to normalize this direction?

True or false: At a regular point of a level curve $f(x,y)=c$, the gradient $\nabla f$ is perpendicular to the curve.

For $f(x,y)=x^2+xy$, which unit vector gives the direction of steepest ascent at $(1,2)$?

For $f(x,y)=x^2-y^2$, how does the second derivative test classify the critical point at $(0,0)$?

In the one-constraint method for optimizing $f$ subject to $g=c$, what is the conventional name of the scalar $\lambda$ multiplying $\nabla g$ in $\nabla f=\lambda\nabla g$?

The surface is $z=f(x,y)=x^2+xy$. Which equation is the tangent plane at the point corresponding to $(x,y)=(1,2)$?