let $f(x)=6x^{4}+2x^{3}$. select all intervals where $f(x)$ is increasing,\n$(-\\frac{1}{4},0)$\n$(-\\infty,\…

let $f(x)=6x^{4}+2x^{3}$. select all intervals where $f(x)$ is increasing,\n$(-\\frac{1}{4},0)$\n$(-\\infty,\\frac{-1}{3})$\n$(0,\\infty)$\n$(\\frac{-1}{3},\\frac{-1}{4})$

let $f(x)=6x^{4}+2x^{3}$. select all intervals where $f(x)$ is increasing,\n$(-\\frac{1}{4},0)$\n$(-\\infty,\\frac{-1}{3})$\n$(0,\\infty)$\n$(\\frac{-1}{3},\\frac{-1}{4})$

Answer

Explanation:

Step1: Find the derivative

The derivative of (f(x)=6x^{4}+2x^{3}) is (f^\prime(x)=24x^{3}+6x^{2}=6x^{2}(4x + 1))

Step2: Determine where (f^\prime(x)>0)

Since (6x^{2}\geq0) for all (x\in R), the sign of (f^\prime(x)) is determined by (4x + 1). Set (4x+1>0), we get (x>-\frac{1}{4}). When (x = 0), (f^\prime(0)=0). For (x>0), (f^\prime(x)=6x^{2}(4x + 1)>0) (because (x^{2}>0) and (4x + 1>1)). For (x\in(-\frac{1}{4},0)), (x^{2}>0) and (4x + 1>0), so (f^\prime(x)>0)

Answer:

((-\frac{1}{4},0)), ((0,\infty))