Question

find all intervals on which ( f(x) ) is increasing.

( f(x)=-3 x^{3}-45 x^{2}-81 x+4 )

Explanation:

Step1: Find the derivative of \(f(x)\)

Using the power rule \((x^n)^\prime = nx^{n - 1}\), for \(f(x)=-3x^{3}-45x^{2}-81x + 4\), we have \(f^\prime(x)=-9x^{2}-90x - 81\). Factor out \(-9\): \(f^\prime(x)=-9(x^{2}+10x + 9)=-9(x + 1)(x + 9)\).

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

Set \(f^\prime(x)>0\), so \(-9(x + 1)(x + 9)>0\). Divide both sides by \(-9\) (and reverse the inequality sign): \((x + 1)(x + 9)<0\).
The roots of the quadratic equation \((x + 1)(x + 9)=0\) are \(x=-9\) and \(x=-1\).
We can use a sign - chart or test intervals.
Test the interval \((-\infty,-9)\): Let \(x=-10\), then \((-10 + 1)(-10+9)=(-9)\times(-1)=9>0\).
Test the interval \((-9,-1)\): Let \(x=-5\), then \((-5 + 1)(-5 + 9)=(-4)\times4=-16<0\).
Test the interval \((-1,\infty)\): Let \(x=0\), then \((0 + 1)(0 + 9)=9>0\).

Answer:

The function \(f(x)\) is increasing on the interval \((-9,-1)\).