Question

independent practice
determine the intervals of decrease of the polynomial function.
f(x)= - 3x^{4}-16x^{3}+6x^{2}+48x + 7
(-22,167)∪(42,∞)
(-∞,-4)∪(-1,1)
(-4,-1)∪(1,∞)
(-∞,167)∪(-22,42)

Explanation:

Step1: Find the derivative

$f'(x)=-12x^{3}-48x^{2}+12x + 48$

Step2: Factor the derivative

$f'(x)=-12(x^{3}+4x^{2}-x - 4)=-12[x^{2}(x + 4)-(x + 4)]=-12(x + 4)(x^{2}-1)=-12(x + 4)(x - 1)(x+1)$

Step3: Determine critical points

Set $f'(x)=0$, then $x=-4,x = 1,x=-1$

Step4: Test intervals

Test the intervals $(-\infty,-4),(-4,-1),(-1,1),(1,\infty)$ using test - points.
For $x<-4$, let $x=-5$, $f'(-5)=-12(-5 + 4)(-5 - 1)(-5+1)<0$
For $-40$
For $-10$
For $x>1$, let $x = 2$, $f'(2)=-12(2 + 4)(2 - 1)(2+1)<0$
The function is decreasing when $f'(x)<0$.

Answer:

$(-\infty,-4)\cup(1,\infty)$