Multi-variable calculus

Question

Consider the function $f : \mathbb{R}^3 \to \mathbb{R}$, defined by $$f(x, y, z) = x + y + z − e^xyz.$$ ($a$) Let $S$ be the level surface of $f$ through the point $P = (0, 0, 1)$. Find the tangent plane to $S$ at $P$. ($b$) By the implicit function theorem, we can describe $S$ by $z = h(x, y)$ near $P$. Calculate $h(0, 0)$.

Answer 1

The tangent plane is given by $f_x(0,0,1)(x-x_0)+f_y(0,0,1)(y-y_0)+f_z(0,0,1)(z-z_0)=0$ with the given values we get $x+0y+1(z-1)=0\rightarrow z=1-x$ so now we have $h(0,0)$ as 1