Prove the monotonic nature of the reciprocal function.

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Prove the monotonic nature of the reciprocal function. (i) Let

0<x_{1}<x_{2}

. Prove that

\dfrac{1}{x_{1}} > \dfrac{1}{x_{2}}

. (ii) Let

x_{1}<x_{2}<0

. Prove that

\dfrac{1}{x_{1}} < \dfrac{1}{x_{2}}

. (iii) Using the results of parts (i) and (ii), explain why the function

y=\dfrac{1}{x}

is decreasing on

(0,\infty)

but increasing on

(-\infty,0)