question\ngiven the function $f(x)=-x^{3}+12x^{2}-36x$, determine all intervals on which $f$ is increasing.
question\ngiven the function $f(x)=-x^{3}+12x^{2}-36x$, determine all intervals on which $f$ is increasing.
Answer
Explanation:
Step1: Find the first - derivative
Differentiate $f(x)=-x^{3}+12x^{2}-36x$ using the power rule. If $y = ax^{n}$, then $y'=nax^{n - 1}$. $f'(x)=-3x^{2}+24x - 36$
Step2: Find the second - derivative
Differentiate $f'(x)$ to get $f''(x)$. $f''(x)=-6x + 24$
Step3: Find where $f''(x)>0$
Set $f''(x)>0$ and solve for $x$. $-6x + 24>0$ $-6x>-24$ $x < 4$
Answer:
$(-\infty,4)$