Determine the relative maximum and minimum on the graph

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SUMMARY

The discussion focuses on determining critical points, intervals of increasing and decreasing functions, and identifying absolute maxima and minima for a function f based on its derivative graph. The critical points identified are at x = 4.5 and x = 13. The intervals of increase are [0, 4.5] and (10, 13], while the intervals of decrease are [4.5, 10) and [13, -∞). The absolute maximum occurs at x = 13, and the absolute minimum is at x = 10.

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  • Understanding of calculus concepts such as critical points and derivatives
  • Knowledge of interval notation and its application in function analysis
  • Familiarity with the concepts of increasing and decreasing functions
  • Ability to interpret graphical representations of functions and their derivatives
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  • Study the properties of derivatives to identify critical points more effectively
  • Learn about the First Derivative Test for determining local maxima and minima
  • Explore the concept of continuity and its impact on function behavior
  • Practice analyzing various functions using graphical methods and calculus techniques
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jaychay
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Given that f is the function on (−∞, ∞) and the graph is the derivative of f

1.) Find the critical point on the graph ?
2.) Find the interval of the increasing function on the graph ?
3.) Find the interval of the decreasing function on the graph ?
4.) Find the point which is the absolute maximum on the graph ?
5.) Find the point which is the absolute minimum on the graph ?
func.png
 
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jaychay said:
Given that f is the function on (−∞, ∞) and the graph is the derivative of f

1.) Find the critical point on the graph ?
2.) Find the interval of the increasing function on the graph ?
3.) Find the interval of the decreasing function on the graph ?
4.) Find the point which is the absolute maximum on the graph ?
5.) Find the point which is the absolute minimum on the graph ?View attachment 10704
Do the graph have the relative maximum and relative minimum ?
 
Let's add the graph of $\color{red}f(x)$ at an arbitrary level. That is, let's pick $\color{red}f(0)=0$.
And let's assume that $\color{red}f(x)$ is continuous at $x=10$.

\begin{tikzpicture}[
declare function={
df1(\x) = 2*cos(3/11*360)-2*cos((\x+1)/11*360);
df2(\x) = 1.25-5/16*(\x-13)^2;
f1(\x) = 2*cos(3/11*360)*\x-2*(sin((\x+1)/11*360) - sin(1/11*360))*11/(2*pi);
f2(\x) = 1.25*(\x-10)-5/16*((\x-13)^3 + 27)/3+f1(10);
}]
%\draw[help lines] (-1,-3) grid (16,4);
\draw[-latex] (-1,0) -- (16,0);
\draw[-latex] (0,-3) -- (0,4);
\draw foreach \i in {1,...,15} { (\i,0.1) -- (\i,-0.1) node[below] {$\i$} };
% \draw foreach \i in {-2,...,2} { (0.1,\i) -- (-0.1,\i) node[ left ] {$\i$} };
\draw[domain=-1:10, variable=\x, thick, smooth] plot ({\x}, {df1(\x)}) (3, {df1(3)}) node[above left] {$f'(x)$};
\draw[domain=10:16, variable=\x, thick, smooth] plot ({\x}, {df2(\x)});
\filldraw[fill=black!5, thick] (10,{df1(10)}) circle (0.05) (10,{df2(10)}) circle (0.05);
\draw[domain=-1:10, variable=\x, red, thick, smooth] plot ({\x}, {f1(\x)}) (9, {f1(9)}) node[above right] {$f(x)$};;
\draw[domain=10:16, variable=\x, red, thick, smooth] plot ({\x}, {f2(\x)});
\end{tikzpicture}

Can we find those points and intervals now?
 
Klaas van Aarsen said:
Let's add the graph of $\color{red}f(x)$ at an arbitrary level. That is, let's pick $\color{red}f(0)=0$.
And let's assume that $\color{red}f(x)$ is continuous at $x=10$.

\begin{tikzpicture}[
declare function={
df1(\x) = 2*cos(3/11*360)-2*cos((\x+1)/11*360);
df2(\x) = 1.25-5/16*(\x-13)^2;
f1(\x) = 2*cos(3/11*360)*\x-2*(sin((\x+1)/11*360) - sin(1/11*360))*11/(2*pi);
f2(\x) = 1.25*(\x-10)-5/16*((\x-13)^3 + 27)/3+f1(10);
}]
%\draw[help lines] (-1,-3) grid (16,4);
\draw[-latex] (-1,0) -- (16,0);
\draw[-latex] (0,-3) -- (0,4);
\draw foreach \i in {1,...,15} { (\i,0.1) -- (\i,-0.1) node[below] {$\i$} };
% \draw foreach \i in {-2,...,2} { (0.1,\i) -- (-0.1,\i) node[ left ] {$\i$} };
\draw[domain=-1:10, variable=\x, thick, smooth] plot ({\x}, {df1(\x)}) (3, {df1(3)}) node[above left] {$f'(x)$};
\draw[domain=10:16, variable=\x, thick, smooth] plot ({\x}, {df2(\x)});
\filldraw[fill=black!5, thick] (10,{df1(10)}) circle (0.05) (10,{df2(10)}) circle (0.05);
\draw[domain=-1:10, variable=\x, red, thick, smooth] plot ({\x}, {f1(\x)}) (9, {f1(9)}) node[above right] {$f(x)$};;
\draw[domain=10:16, variable=\x, red, thick, smooth] plot ({\x}, {f2(\x)});
\end{tikzpicture}

Can we find those points and intervals now?
My answers for all of the question that I have done on my own is
  • 1.) 4.5,13
  • 2.) [0,4.5] , (10,13]
  • 3.) [4.5,10), [13, - ∞)
  • 4.) 4.5,13
  • 5.) 10
I want to check that I am doing it correct or not because I am not good at discontinuous graph
Can you please help me
Thank you in advice
 
jaychay said:
Given that f is the function on (−∞, ∞) and the graph is the derivative of f

1.) Find the critical point on the graph ?
  • 1.) 4.5,13

A critical point is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero.

That's not the case for x=4.5 or x=13 is it?
For which x-values is $f'(x)$ either zero or undefined?

jaychay said:
2.) Find the interval of the increasing function on the graph ?
3.) Find the interval of the decreasing function on the graph ?
  • 2.) [0,4.5] , (10,13]
  • 3.) [4.5,10), [13, - ∞)

A function is increasing if its derivative is greater than zero.
Where is $f'(x)$ greater than zero?

We can also look at the graph for $f(x)$ that I added.
Where is that graph increasing?

jaychay said:
4.) Find the point which is the absolute maximum on the graph ?
5.) Find the point which is the absolute minimum on the graph ?
  • 4.) 4.5,13
  • 5.) 10

We have relative maxima and minima if $f'(x)=0$.
For which x-values is $f'(x)$ equal to zero?
Can we tell whether they correspond to maxima or minima?
 
Klaas van Aarsen said:
A critical point is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero.

That's not the case for x=4.5 or x=13 is it?
For which x-values is $f'(x)$ either zero or undefined?
A function is increasing if its derivative is greater than zero.
Where is $f'(x)$ greater than zero?

We can also look at the graph for $f(x)$ that I added.
Where is that graph increasing?
We have relative maxima and minima if $f'(x)=0$.
For which x-values is $f'(x)$ equal to zero?
Can we tell whether they correspond to maxima or minima?
Thank you very much
 

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