So I am completely lost right now and don't exactly know what my professor is asking me.
For Problem 3)
I figured if I am applying Directional Derivative twice on a function f, then:
$ D_u(D_uf) = f_x{_x} (cos^2\theta) + f_y{_y} (sin^2\theta) $
For problem 4, questions of this type has been the most difficult for me to understand.
So here is how I tried to wrap my head around this problem and I am most likely completely wrong.
I just tried to imagine a tangent line for each plot and depending on the direction from the origin, I figured if the tangent line is negative, I can see in which direction it is decreasing altitude and therefore, determine a relative min or max.
For instance, we see that from 2 to 3, the tangent line is negative so there is a downward hill going from 2 to 3. Also, the altitude is decreasing from 3 to 4 even thought it is in the opposite direction of 2 so its like spiraling down?
I don't know. Someone please save me
EDIT:
Wait for problem 3 I think I figured it out just now:
Using the definition of Directional Derivatives:
Du(Duf) is Simply multiplying Du by Du and ultimately we get:
fxxcos^2o * fyy (sin^2o) - [fxy]^2?
$$D_{\mathbf u}f=\mathbf u \bullet \nabla f=[3/5,4/5] \bullet [f_x,f_y]=3f_x/5+4f_y/5$$ $$D_{\mathbf u}(D_{\mathbf u}f)=[3/5,4/5] \bullet \nabla (3f_x/5+4f_y/5)$$ $$=[3/5,4/5] \bullet [3f_{xx}/5+4f_{yx}/5,3f_{xy}/5+4f_{yx}/5]$$ $$=9f_{xx}/25+12f_{yx}/25+12f_{xy}/25+16f_{yy}/25$$ $$=\begin{bmatrix}3/5&4/5\end{bmatrix}\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}\begin{bmatrix}3/5\\4/5\end{bmatrix}$$ $$D_{[\cos \theta,\sin \theta]}(D_{[\cos \theta,\sin \theta]}f)=\begin{bmatrix}\cos \theta&\sin \theta\end{bmatrix}\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}\begin{bmatrix}\cos \theta\\\sin \theta\end{bmatrix}$$ In general, $$D_{\mathbf u}(D_{\mathbf u}f)=\mathbf u \text{ Hess(f) }\mathbf u^T.$$ Note that, by Clairaut's theorem the Hessian matrix is symmetric, so the quadratic form above is positive definite iff $f_{xx}>0$ and the determinant $f_{xx}f_{yy}-f_{xy}f_{yx}>0$