First slide

Monotonicity

Question

$The function y = \frac{x}{1 + x^{2}} decreases in the interval$

Easy

A

$(- 1, 1)$
B

$[1, \infty)$
C

$(- \infty, - 1]$
D

$(- \infty, \infty)$

Solution

$y = \frac{x}{1 + x^{2}} \frac{dy}{dx} = \frac{1 + x^{2} - 2 x^{2}}{{(1 + x^{2})}^{2}} = \frac{1 - x^{2}}{{(1 + x^{2})}^{2}} for decreasing 1 - x^{2} < 0 \Rightarrow x \in (- 1, - 1)$

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