find the intervals cor which the function h(x) = x^3 −3x is a) strickly increasing b) strickly decreasing


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Question Number 75100 by Rio Michael last updated on 07/Dec/19

$f i n d t h e i n t e r v a l s c o r w h i c h t h e f u n c t i o n$ $h (x) = x^{3} - 3 x i s$ $a) s t r i c k l y i n c r e a s i n g$ $b) s t r i c k l y d e c r e a s i n g$
	Answered by mr W last updated on 07/Dec/19

	$h^{'} (x) = 3 x^{2} - 3 = 3 (x^{2} - 1)$ $(a)$ $h^{'} (x) = 3 (x^{2} - 1) > 0$ $\Rightarrow x^{2} > 1$ $\Rightarrow x < - 1 o r x > 1$ $(b)$ $h^{'} (x) = 3 (x^{2} - 1) < 0$ $\Rightarrow x^{2} < 1$ $\Rightarrow - 1 < x < 1$
	Answered by Kunal12588 last updated on 07/Dec/19

	$f (x) = x^{3} - 3 x$ $y = x^{3} - 3 x$ $\Rightarrow \frac{d y}{d x} = 3 x^{2} - 3 = 3 (x - 1) (x + 1)$ $p u t t i n g \frac{d y}{d x} = 0 g i v e s x = \pm 1$ $w h i c h d i v i d e s t h e n u m b e r l i n e i n t o 3 i n t e r v a l s$ $(- \infty, - 1), (- 1, 1) a n d (1, \infty)$ $i n t e r v a l ∣ s i g n o f f^{'} (x) ∣ n a t u r e o f f (x)$ $(- \infty, - 1) ∣ (-) (-) ∣ s t r i c t l y i n c r e a s i n g$ $(- 1, 1) ∣ (-) (+) ∣ s t r i c t l y d e c r e a s i n g$ $(1, \infty) ∣ (+) (+) ∣ s t r i c t l y i n c r e a s i n g$ $∴ f (x) i s s t r i c t l y i n c r e a s i n g i n (- \infty, - 1) \cup (1, \infty)$ $f (x) i s s t r i c t l y d e c r e a s i n g i n (- 1, 1)$
	Commented by Rio Michael last updated on 07/Dec/19

	$w o w s i r i a p p r e c i a t e$
	Commented by peter frank last updated on 07/Dec/19

	$g o o d$
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