Which of the following statements is always true?

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If $f$ and $g$ are positive functions and $f$ is decreasing and g is increasing, then $f / g$ is a decreasing function.

If f(x) is increasing, then $\frac{1}{f (x)}$ is also increasing.

If $f$ and $g$ are positive functions and $f$ is increasing and g is decreasing, then $f / g$ is a decreasing function.

If $f (x)$ is increasing, then $f^{- 1} (x)$ is decreasing.

answer is D.

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Detailed Solution

If $f (x)$ increases, then $f^{- 1} (x)$ increases.

If $f (x)$ increases, then $f' (x) > 0$

$\Rightarrow$ $\frac{d}{d x} (\frac{1}{f (x)}) = - \frac{f^{'} (x)}{f^{2} (x)} < 0$

Thus, $\frac{1}{f (x)}$ decreases. $\frac{d}{d x} (\frac{f}{g}) = \frac{f^{'} g - f g^{'}}{g^{2}}$

if $f$ and g are $+ v e$ functions and $f' < 0 a n d g' > 0,$ then $\frac{d}{d x} (\frac{f}{g}) < 0$

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