(4) Fix a € C\{0,1}, and let F = {fe Hol(D) | f(D) C C\{0,1}, f(0) = a}. PƒeƑ |ƒ'(0)| < ∞. [Hint: lift, and use the Schwarz.] Prove that sup

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2. Let f : R" → R be a continuously differentiable function, while a E R" is its non-stationary point (i.e., Vf(x) # 0). Moreover, let d* be the vector defined by Vf(x) ||Vf(x)||' d* = - while d is any unit vector in R" different from d* (i.e., d e R", ||d|| = 1, d + d*). Show that there exist real numbers d E (0, 00), ɛ E (0, 0) such that %3D f(x + td*) – f(x + td) 0. (ii) For t z 0, approximate f(x+td*) and f(x+td) using the first-order Taylor polynomial of f(x) at x.

13. Find an equation for the tangent plane of the graph of f at the point (xo, yo, f(xo, yo)) for: (a) ƒ: R²2 → R, (x, y) → x − y +2, (xo, yo) = (1, 1) (b) (c) ƒ: R² → R, (x, y) ↔ x² + 4y², (xo, yo) = (2, -1) f: ƒ: R² R, (x, y) → xy, → (xo, yo) = (-1, −1) (d) f(x, y) = log (x + y) + x cos y + arctan(x + y), (xo, yo) = (1, 0) -2 (e) f(x, y) = √√√x² + y², (f) f(x, y) = xy, (xo, yo) = (1, 1) (xo, yo) = (2, 1)