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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=f(x)} .

Foundations:
Recall:
1. $f(x)$  is increasing when Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 $f''(x)=0.$

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 (-\infty ,0),(0,4),(4,\infty ).} $0=3x(x-4).$
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 $(-\infty ,0)\cup (4,\infty ),$  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 $f(0)=5$ 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 $f''(x)=6x-12.$
We set $f''(x)=0.$
So, we have $0=6x-12.$ Hence, $x=2.$
This value breaks up the number line into two intervals: $(-\infty ,2),(2,\infty ).$

Step 2:
Again, we use test points in these two intervals.
For $x=0,$  we have $f''(x)=-12<0.$
For $x=3,$  we have $f''(x)=6>0.$
Thus, $f(x)$  is concave up on the interval $(2,\infty ),$ and concave down on the interval $(-\infty ,2).$

(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  $(2,-11).$

(e)

Final Answer:
(a) $f(x)$  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) The local maximum value is $f(0)=5,$  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) 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) 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 in (e).

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