Question

question
given the function $f(x) = x^3 + 18x^2 + 81x$, determine all intervals on which $f$ is decreasing.

Explanation:

Step1: Find first derivative $f'(x)$

Differentiate $f(x)=x^3+18x^2+81x$ using power rule:
$f'(x) = 3x^2 + 36x + 81$

Step2: Find second derivative $f''(x)$

Differentiate $f'(x)$ to analyze its monotonicity:
$f''(x) = 6x + 36$

Step3: Find critical point of $f'(x)$

Set $f''(x)=0$ and solve for $x$:
$6x + 36 = 0 \implies x = -6$

Step4: Test intervals for $f''(x)$ sign

For $x < -6$, $f''(x) < 0$; for $x > -6$, $f''(x) > 0$. $f'(x)$ decreases when $f''(x) < 0$.

Answer:

$(-\infty, -6)$