1) T F The function f(x, y) = x^6 + y^6 has exactly one minimum.
2) T F The gradient ∇f(1, 1, 1) is tangent to the surface x^2 + y^2 − z^2 = 1.
3) T F The directional derivative D~vf(0, 0) is positive if the unit
4) T F If fx(0, 0) = fy(0, 0) = 0 and fxx(0, 0) > 0, fyy(0, 0) > 0, fxy(0, 0) < 0, then (0, 0) is a local minimum of f.
5) T F If fx(0, 0) = fy(0, 0) = 0 and fxx > 0, fyy(0, 0) = 0, fxy(0, 0) < 0, then (0, 0) is a saddle point.
6) T F The chain rule tells that d dtf(~r(t)) = ∇f(~r(t)) · ∇f(~r 0 (t)) for any function f(x, y, z) and any curve ~r(t) = [x(t), y(t), z(t)]T
7) T F The point (0, 1) is a maximum of f(x, y) = x + y under the constraint g(x, y) = x = 1.
8) T F The function u(x, y) = √ x^2 + y^2 solves the partial differential equation u^2 x + u^2 y = 1
9) T F Let f(x, y) = x^3 y^2 . At (0, 0) and every direction ~v we have D~vf(0, 0) = 0.
10) T F The identity fxyx = fyxy holds for all smooth functions f(x, y).
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