find intervals where this function is increasing, decreasing, or constant.\nf(x)=x^{3}-x^{2}-x\nincreasing…

Explanation:

Step1: Find the derivative

Differentiate $f(x)=x^{3}-x^{2}-x$ using the power - rule. The derivative $f'(x)=3x^{2}-2x - 1$.

Step2: Set the derivative equal to zero

Solve $3x^{2}-2x - 1 = 0$. Factor the quadratic equation: $3x^{2}-2x - 1=(3x + 1)(x - 1)=0$. Then $3x+1 = 0$ gives $x=-\frac{1}{3}\approx - 0.33$ and $x - 1=0$ gives $x = 1$.

Step3: Test intervals

Choose test points in the intervals $(-\infty,-\frac{1}{3})$, $(-\frac{1}{3},1)$ and $(1,\infty)$. For the interval $(-\infty,-\frac{1}{3})$, let $x=-1$. Then $f'(-1)=3+2 - 1=4>0$, so the function is increasing on $(-\infty,-\frac{1}{3})$. For the interval $(-\frac{1}{3},1)$, let $x = 0$. Then $f'(0)=-1<0$, so the function is decreasing on $(-\frac{1}{3},1)$. For the interval $(1,\infty)$, let $x = 2$. Then $f'(2)=12-4 - 1 = 7>0$, so the function is increasing on $(1,\infty)$.

Answer:

decreasing from -0.33 to 1, otherwise increasing