A Decreasing Function?

Calculus Level 2

True or False?

Consider the function $f(x) = \frac{1}{x}$ . Since $f'(x) = -\frac{1}{x^{2}} < 0$ , $f(x)$ is a decreasing function in its domain $(-\infty, 0) \cup (0, \infty)$ .

True False

1 solution

Yash Dev Lamba
Mar 2, 2016

Function is decreasing in each branch only.

For example,

$2>1$ $\implies$ $f(2)<f(1)$ $\implies$ decreasing function

$1>-1$ $\implies$ $f(1)>f(-1)$ $\implies$ not decreasing function

Therefore, given function is not decreasing in $(-\infty, 0) \cup (0, \infty)$ , indeed it is decreasing in $(-\infty, 0)$ & $(0, \infty)$ indiviually

Another way of putting it is, as f is not continuous at zero, the derivative test doesn't apply.

A Former Brilliant Member - 5 years, 3 months ago

A Decreasing Function?

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