91Ó°ÊÓ

Explain what is wrong with the statement. A function \(f\) has gradient grad \(f(0,0)=7\)

Give an example of: A unit vector \(\vec{u}\) such that \(f_{\vec{u}}(0,0)<0,\) given that \(f_{x}(0,0)=2\) and \(f_{y}(0,0)=3\)

True or false? Give reasons for your answer. The gradient vector grad \(f(a, b)\) is a vector in 3 -space.

True or false? Give reasons for your answer. $$\operatorname{grad}(f g)=(\operatorname{grad} f) \cdot(\operatorname{grad} g)$$

True or false? Give reasons for your answer. If you know the gradient vector of \(f\) at \((a, b)\) then you can find the directional derivative \(f_{\vec{u}}(a, b)\) for any unit vector \(\vec{u}\)

True or false? Give reasons for your answer. If you know the directional derivative \(f_{\vec{a}}(a, b)\) for all unit vectors \(\vec{u}\) then you can find the gradient vector of \(f\) at \((a, b)\)

True or false? Give reasons for your answer. The directional derivative \(f_{\vec{u}}(a, b)\) is parallel to \(\vec{u}\)

True or false? Give reasons for your answer. If grad \(f(1,2)=\vec{i},\) then \(f\) decreases in the \(-\vec{i}\) direction at (1,2)

True or false? Give reasons for your answer. At the point \((3,0),\) the function \(g(x, y)=x^{2}+y^{2}\) has the same maximal rate of increase as that of the function \(h(x, y)=2 x y\)

True or false? Give reasons for your answer. If \(f(x, y)=e^{x+y},\) then the directional derivative in any direction \(\vec{u}\) (with \(\|\vec{u}\|=1\) ) at the point (0,0) is always less than or equal to \(\sqrt{2}\)