True or false, when … is any smooth function: (a) There is…

True or false, when $f(x,y)$ is any smooth function:
(a) There is a direction $\mathbf{u}$ at $P$ in which $D_{\mathbf{u}}f=0$.
(b) There is a direction $\mathbf{u}$ in which $D_{\mathbf{u}}f=\text{grad }f$.
(c) There is a direction $\mathbf{u}$ in which $D_{\mathbf{u}}f=1$.
(d) The gradient of $f(x)g(x)$ equals $g\text{grad }f+f\text{grad }g$.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §13.4 Directional Derivatives and Gradients, #11