Question
Find the intervals in which the following functions are increasing or decreasing.
f(x) = 2x3 + 9x2 + 12x + 20
f(x) = 2x3 + 9x2 + 12x + 20
✓
Answer
f(x) = 2x3 + 9x2 + 12x + 20
f'(x) = 6x2 + 18x + 12
= 6(x2 + 3x + 2)
= 6(x + 1)(x + 2)
For f(x) to be increasing, we must have
f'(x) > 0
⇒ 6(x + 1)(x + 2) > 0
⇒ (x + 1)(x + 2) > 0
[Since, 6 > 0, 6(x + 1)(x + 2) > 0 ⇒ (x + 1)(x + 2) > 0]
⇒ x < -2 or x > -1
$\Rightarrow\text{x}\in(-\infty,-2)\cup(-1,\infty)$
So, f(x) is increasing on $\text{x}\in(-\infty,-2)\cup(-1,\infty).$
For f(x) to be decreasing, we must have,
f'(x) < 0
⇒ 6(x + 1)(x + 2) < 0
⇒ (x + 1)(x + 2) < 0
[Since, 6 > 0, 6(x + 1)(x + 2) < 0 ⇒ (x + 1)(x + 2) < 0]
⇒ -2 < x < -1
$\Rightarrow\text{x}\in(-2,-1)$
So, f(x) is decreasing on $\text{x}\in(-2,-1).$
f'(x) = 6x2 + 18x + 12
= 6(x2 + 3x + 2)
= 6(x + 1)(x + 2)
For f(x) to be increasing, we must have
f'(x) > 0
⇒ 6(x + 1)(x + 2) > 0
⇒ (x + 1)(x + 2) > 0
[Since, 6 > 0, 6(x + 1)(x + 2) > 0 ⇒ (x + 1)(x + 2) > 0]
⇒ x < -2 or x > -1
$\Rightarrow\text{x}\in(-\infty,-2)\cup(-1,\infty)$
So, f(x) is increasing on $\text{x}\in(-\infty,-2)\cup(-1,\infty).$
For f(x) to be decreasing, we must have,
f'(x) < 0
⇒ 6(x + 1)(x + 2) < 0
⇒ (x + 1)(x + 2) < 0
[Since, 6 > 0, 6(x + 1)(x + 2) < 0 ⇒ (x + 1)(x + 2) < 0]
⇒ -2 < x < -1
$\Rightarrow\text{x}\in(-2,-1)$
So, f(x) is decreasing on $\text{x}\in(-2,-1).$
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