Which of the following best explains why the curve has no horizontal tangent lines for ?

MATHEMATICS · Single correct · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

Which of the following best explains why the curve

y=\ln x

has no horizontal tangent lines for

x>0

? (a) Since

\ln x

is always increasing for

x>0

, every tangent line must have the same positive slope

1

, so none are horizontal. (b) Because

y''=-\dfrac{1}{x^{2}}

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

\dfrac{d}{dx}\ln x = \dfrac1x

, which is never zero when

x>0

. (d) A horizontal tangent occurs where the function value is zero, and

\ln x=0

only when

x=1

, so the curve has no horizontal tangent elsewhere on

x>0