009A Sample Final 1, Problem 9

Given the function $f(x)=x^{3}-6x^{2}+5$,

a) Find the intervals in which the function increases or decreases.

b) Find the local maximum and local minimum values.

c) Find the intervals in which the function concaves upward or concaves downward.

d) Find the inflection point(s).

e) Use the above information (a) to (d) to sketch the graph of $y=f(x)$.

Foundations:

Solution:

(a)

Step 1:
We start by taking the derivative of $f(x)$. We have $f'(x)=3x^{2}-12x$.
Now, we set $f'(x)=0$. So, we have $0=3x(x-4)$.
Hence, we have $x=0$ and $x=4$.
So, these values of $x$ break up the number line into 3 intervals: $(-\infty ,0),(0,4),(4,\infty )$.

Step 2:
To check whether the function is increasing or decreasing in these intervals, we use testpoints.
For $x=-1$, $f'(x)=15>0$.
For $x=1$, $f'(x)=-9<0$.
For $x=5$, $f'(x)=15>0$.
Thus, $f(x)$ is increasing on $(-\infty ,0),(4,\infty )$ and decreasing on $(0,4)$.

(b)

Step 1:

Step 2:

Step 3:

(c)

Step 1:

Step 2:

(d)

Step 1:

Step 2:

(e)

Step 1:

Step 2:

Final Answer:
(a) $f(x)$ is increasing on $(-\infty ,0),(4,\infty )$ and decreasing on $(0,4)$.
(b)
(c)
(d)
(e)

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