Question

When the curves $y = \log_{10} x$ and $y = x^{-1}$ are drawn in the $x$-$y$ plane, how many times do they intersect for values $x \ge 1$?

Accepted Answer

B. At $x = 1$, $y_1 = \log_{10}(1) = 0$ and $y_2 = 1^{-1} = 1$. So $y_1 < y_2$. As $x 	o \infty$, $y_1 = \log_{10} x 	o \infty$ while $y_2 = 1/x 	o 0$. So $y_1 > y_2$. Since both functions are continuous for $x \ge 1$, $y_1$ is strictly increasing and $y_2$ is strictly decreasing, they intersect exactly once in the interval $(1, \infty)$.

Intersection of Logarithmic and Reciprocal Curves

Explanation

Practise this under exam conditions