2. Let $f(x, y, z)=x e^{x y z}-arcsin left(2 x+y^{2} ight)-z y+2$ be a function of...

  

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2. Let $f(x, y, z)=x e^{x y z}-\arcsin \left(2 x+y^{2} ight)-z y+2$ be a function of three variables. (a) Find the domain of definition of the above function and decide if it is closed or open, bounded or unbounded set. (b) Write the equation of the tangentplane to the surface $x e^{x y z}- \arcsin \left(2 x+y^{2} ight)-$ $z y+2=0$ at the point $(-2,2,0)$. (c) Calculate the smallest possible value of the directional derivative $D_{\vec {u}} f(-2,2,0)$ if $\vec {u} |=1$. In which direction is this value attained? $(9 p) $ CS.VS.1651| 2. Let $f(x, y, z)=x e^{x y z}-\arcsin \left(2 x+y^{2} ight)-z y+2$ be a function of three variables. (a) Find the domain of definition of the above function and decide if it is closed or open, bounded or unbounded set. (b) Write the equation of the tangentplane to the surface $x e^{x y z}- \arcsin \left(2 x+y^{2} ight)-$ $z y+2=0$ at the point $(-2,2,0)$. (c) Calculate the smallest possible value of the directional derivative $D_{\vec {u}} f(-2,2,0)$ if $\vec {u} |=1$. In which direction is this value attained? $(9 p) $ CS.VS.1651|