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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=3x^2-12x} .
Now, we set $f'(x)=0$. So, we have Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-\infty ,0),(0,4),(4,\infty )} .

Step 2:
To check whether the function is increasing or decreasing in these intervals, we use testpoints.
For $x=-1$, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(x)=15>0} .
For $x=1$, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(x)=-9<0} .
For $x=5$, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(x)=15>0} .
Thus, $f(x)$ is increasing on $(-\infty ,0),(4,\infty )$ and decreasing on Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (0,4)} .

(b)

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Step 2:

Step 3:

(c)

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Step 2:

(d)

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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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