Which of the following best explains why the curve y=lnx has no horizontal tangent lines for x>0?
a) Since lnx is always increasing for x>0, every tangent line must have the same positive slope 1, so none are horizontal.
b) Because y′′=−x21 is negative for all x>0, the graph is always concave down, and a curve that is always concave down cannot have a horizontal tangent.
c) The derivative dxdlnx=x1, which is never zero when x>0.
d) A horizontal tangent occurs where the function value is zero, and lnx=0 only when x=1, so the curve has no horizontal tangent elsewhere on x>0.