find-the-intervals-cor-which-the-function-h-x-x-3-3x-is-a-strickly-increasing-b-strickly-decreasing-

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