Suppose ||$\nabla$f(a, b, c)||=19.Is it possible to choose a direction from (a, b, c)so that $f _ { \vec { u } }$ in that direction is -19?

Question

Quizplus · Accepted Answer

The directional derivative of a function in the direction of a unit vector $\vec{u}$ is given by the dot product of the gradient of the function and the unit vector $\vec{u}$. Since the magnitude of the gradient is 19, by choosing a direction opposite to the gradient (i.e., $\vec{u}$ is in the opposite direction of the gradient), the directional derivative $f_{\vec{u}}$ can indeed be -19, as it would be the product of the magnitude of the gradient and the cosine of 180 degrees (which is -1).

Suppose || ∇\nabla∇ F(a, B, C)||=19 fu⃗f _ { \vec { u } }fu​

Suppose || $\nabla$ F(a, B, C)||=19 $f _ { \vec { u } }$