009A Sample Final 1, Problem 9 Detailed Solution

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  $y=f(x)$.

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Background Information:
Recall:
1. $f(x)$  is increasing when  $f'(x)>0$  and  $f(x)$  is decreasing when  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(x)<0.}
2. The First Derivative Test tells us when we have a local maximum or local minimum.
3. $f(x)$  is concave up when  $f''(x)>0$  and  $f(x)$  is concave down when  $f''(x)<0.$
4. Inflection points occur when  $f''(x)=0$  and the second derivative changes sign.

Solution:

(a)

Step 1:
We start by taking the derivative of  $f(x).$
We have  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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  $x=-1,~f'(x)=15>0.$
For  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=1,~f'(x)=-9<0.}
For  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=5,~f'(x)=15>0.}
Thus,  $f(x)$  is increasing on  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-\infty ,0)\cup (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)

Step 1:
By the First Derivative Test, the local maximum occurs at  $x=0$
and the local minimum occurs at  $x=4.$

Step 2:
So, the local maximum value is  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(0)=5}
and the local minimum value is  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(4)=-27.}

(c)

Step 1:
To find the intervals when the function is concave up or concave down,
we need to find  $f''(x).$
We have  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f''(x)=6x-12.}
We set  $f''(x)=0.$
So, we have
$0=6x-12.$
Hence,  $x=2.$
This value breaks up the number line into two intervals:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-\infty ,2),(2,\infty ).}

Step 2:
Again, we use test points in these two intervals.
For  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=0,}   we have  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f''(x)=-12<0.}
For  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=3,}   we have  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f''(x)=6>0.}
Thus,  $f(x)$  is concave up on the interval  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (2,\infty ),}   and concave down on the interval  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-\infty ,2).}

(d)
Using the information from part (c), there is one inflection point that occurs at  $x=2.$
Now, we have  $f(2)=8-24+5=-11.$
So, the inflection point is  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (2,-11).}

(e)
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Final Answer:
(a)    $f(x)$  is increasing on  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-\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)    The local maximum value is  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(0)=5}   and the local minimum value is  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(4)=-27.}
(c)    $f(x)$  is concave up on the interval  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 (2,\infty)}   and concave down on the interval  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 (-\infty,2).}
(d)     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 (2,-11)}
(e)     See graph above.

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