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:
Recall:
1. $f(x)$  is increasing when  $f'(x)>0$  and  $f(x)$  is decreasing when  $f'(x)<0.$
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  $f''(x)=0.$

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  $(-\infty ,0)\cup (4,\infty ),$  and decreasing on  $(0,4).$

(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  $f(0)=5$  and the local minimum value is  $f(4)=-27.$

(c)

Step 1:
To find the intervals when the function is concave up or concave down, we need to find  $f''(x).$
We have  $f''(x)=6x-12.$
We set  $f''(x)=0.$
So, we have
$0=6x-12.$  Hence,  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 x=2.}
This value breaks up the number line into two intervals:
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),(2,\infty).}

Step 2:
Again, we use test points in these two intervals.
For  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 x=0,}   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)=-12<0.}
For  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 x=3,}   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)=6>0.}
Thus,  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)}   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)
Using the information from part (c), there is one inflection point that occurs at  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 x=2.}
Now, 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(2)=8-24+5=-11.}
So, the inflection point is  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)
Insert sketch here.

Final Answer:
(a)    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)}   is increasing on  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,0),(4,\infty),}   and decreasing on  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 (0,4).}
(b)    The local maximum value is  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(0)=5,}   and the local minimum value is  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(4)=-27.}
(c)    $f(x)$  is concave up on the interval  $(2,\infty ),$  and concave down on the interval  $(-\infty ,2).$
(d)     $(2,-11)$
(e)     See graph above.

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