1. the level curves of z=2x^2+3y^2 are ellipses.
2.if fx(0,0) exists,then g(x)=f(x,0) is continuous at x=0.
3. if lim f(x,y) =L, then lim y→0 f(y,y)=L.
4.if f(x,y)=g(x)h(y),where g and h are continuous for all x and y,respectively,then f is continuous on the whole xy-plane.
5. if f(x,y) and g(x,y) have the same gradient , then they are identical functions.
6.if f is differentiable and ▽f(a,b)=0 ,then the graph of z=f(x,y) has a horizontal tangent plane at (a,b).
7. if ▽f(p0)=0,then f has an extreme value at p0.
- 1 0 年前最佳解答
the level curves are
x^2/(k/2) + y^2/(k/3) =1
which are ellipses.
g'(0)=f_x(0,0) 存在，故 g 在 0 連續
Let G(x,y)=g(x), H(x,y)=h(y)
Then G, H are contiuous, so is their product GH.
thus f=gh=GH is continuous.
f and g differ by a constant
the tangent plane of the graph z=f(x,y) at (a,b) has normal vector equal to
▽f(a,b) + k = k, where k is the upward unit vector in the z axis.
Hence it is horizontal.
7. NO. For example, f(x,y)=xy. ▽f(0,0)=(0,0), but f has a saddle point at the origin.