Question

f(x) = sqrt(x^2 + 0.0001)
At x = 0, which of the statements is true.
a)f is increasing
b)f is discontinuous
c)f has a horizontal tangent
d)f' is undefined

Answer is c but why?

f(x,y) = sqrt(x^2 + y)

g(x,y) = df/dx =

1/(2*sqrt[x^2 + y]) * 2x =

x/sqrt[x^2 + y])

For y>0 this is zero when x = 0.

But for y = 0 you find:

g(x,0) = 1

This means that Lim y --> 0 of g(0,y)=0