Find the intervals in which `f(x)=5+36 x+3x^2-2x^3` is increasing or decreasing.

Explanation

Given as `f(x) = 5 + 36x + 3x^2 – 2x^3`

` f'(x) = (d/dx)(5 + 36x + 3x^2 - 2x^3) `

`⇒ f’(x) = 36 + 6x – 6x^2`
For the function `f(x)` now we have to find critical point, we must have `⇒ f’(x) = 0 ⇒ 36 + 6x – 6x^2 = 0 `

`⇒ 6(–x^2 + x + 6) = 0`

` ⇒ 6(–x^2 + 3x – 2x + 6) = 0`

` ⇒ –x^2 + 3x – 2x + 6 = 0 `

`⇒ x^2 – 3x + 2x – 6 = 0`
`⇒ (x – 3) (x + 2) = 0 `
`⇒ x = 3, – 2`

It is clear, `f’(x) > 0` if ` –2< x < 3` and `f’(x) < 0` if `x < –2` and `x > 3`
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