Local minimum and area of between two points on graph.

Zohaibali

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Hello All dear Fellows..

I need help in finding the local minimum. First i find the derivative of the given function f(x).
Now i don't know how to find the local minimum and the area of between points on the graph.

I have attached my work. Please help in finding the minimum and area between the point f and x=0.
 

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I have moved your question from the first subforum listed on the front page ("News") to a topic-appropriate category. In future, kindly please conform to what is explained in the "Read Before Posting" announcement. Thank you for your consideration.

I need help in finding the local minimum. First i find the derivative of the given function f(x). Now i don't know how to find the local minimum and the area of between points on the graph.

I have attached my work. Please help in finding the minimum and area between the point f and x=0.
Which portion(s) is/are the exercise? Which is/are your work? What do you mean when you say that you "don't know how to find the local minimum and the area...between points on a graph" (by which I assume you mean "the area under a curve, between two given endpoints")? Are you saying that you are needing instruction in calculus techniques? Or that you don't know how to use the specified software? Or something else?

For other readers, the text in the graphics is:



This question is about the function

. . . . .f(x)=x5+6x322x25x3x2+2\displaystyle f(x)\, =\, \dfrac{x^5\, +\, 6x^3\, -\, 22x^2\, -\, 5x}{3x^2\, +\, 2}

Use Maxima to do each of parts (a) through (d).

(a) Plot the graph of f, choosing ranges on the x- and y-axes to make its two stationary points clearly visible.

(b) Find the derivative of f.

(c) Calculate the x- and y-coordinates of the local minimum of f, to three significant figures.

(d) Find the area between the graph of f and the x-axis, from x = 0 to the point where the graph next crosses the x-axis, to three significant figures.




This is the poster's work (I think):

(a)
attachment.php


(or view here)

(b) \(\displaystyle \begin{align} \dfrac{d}{dx}\,\left(\, \dfrac{x^5\, +\, 6x^3\, -\, 22x^2\, -\, 5x}{3x^2\, +\, 2}\,\right)\, &=\, \dfrac{(3x^2\, +\, 2)\,\cdot\, \dfrac{d}{dx}\,(x^5\, +\, 6x^3\, -\, 22x^2\, -\, 5x)\, -\, \left(x^5\, +\, 6x^3\, -\, 22x^2\, -\, 5x\right)\,\cdot\,\dfrac{d}{dx}\, (3x^2\, +\, 2)}{(3x^2\, +\, 2)^2}

\\ \\ &=\, \dfrac{(3x^2\, +\, 2)\, (5x^4\, +\, 18x^2\, -\, 44x\, -\, 5)\, -\, (x^5\, +\, 6x^3\, -\, 22x^2\, -\, 5x)\, (6x)}{9x^4\, +\, 12x^2\, +\, 4}

\\ \\ &=\, \dfrac{(15x^6\, +\, 54x^4\, -\, 132x^3\, -\, 15x^2)\, +\, (10x^4\, +\, 36x^2\, -\, 88x\, -\, 10)\, -\, (6x^6\, +\, 36x^4\, -\, 132x^3\, -\, 30x^2)}{9x^4\, +\, 12x^2\, +\, 4}

\\ \\ &=\, \dfrac{9x^6\, +\, 28x^4\, +\, 51x^2\, -\, 88x\, -\, 10}{9x^4\, +\, 12x^2\, +\, 4}\end{align}\)
 

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This question is about the function

. . . . .f(x)=x5+6x322x25x3x2+2\displaystyle f(x)\, =\, \dfrac{x^5\, +\, 6x^3\, -\, 22x^2\, -\, 5x}{3x^2\, +\, 2}

Use Maxima to do each of parts (a) through (d).

(a) Plot the graph of f, choosing ranges on the x- and y-axes to make its two stationary points clearly visible.

(b) Find the derivative of f.

(c) Calculate the x- and y-coordinates of the local minimum of f, to three significant figures.

(d) Find the area between the graph of f and the x-axis, from x = 0 to the point where the graph next crosses the x-axis, to three significant figures.
\(\displaystyle \mbox{(b) }\,\dfrac{d}{dx}\,\left(\, \dfrac{x^5\, +\, 6x^3\, -\, 22x^2\, -\, 5x}{3x^2\, +\, 2}\,\right)\, =\,...\, =\, \dfrac{9x^6\, +\, 28x^4\, +\, 51x^2\, -\, 88x\, -\, 10}{9x^4\, +\, 12x^2\, +\, 4}\)
Note: I corrected the typoed sign on 51x^2, which should be "plus" rather than "minus" (as in your graphic).

(c) To find the x-coordinate of the local min, set the derivative equal to zero, same as always:

. . . . .9x6+28x4+51x288x109x4+12x2+4=0\displaystyle \dfrac{9x^6\, +\, 28x^4\, +\, 51x^2\, -\, 88x\, -\, 10}{9x^4\, +\, 12x^2\, +\, 4}\, =\, 0

. . . . .9x6+28x4+51x288x10=0\displaystyle 9x^6\, +\, 28x^4\, +\, 51x^2\, -\, 88x\, -\, 10\, =\, 0

Presumably, you're expected to solve this within the specified software package. The solution (being an x-value slightly greater than 2) is the x-coordinate of the local min. I would expect that the software can evaluate f(x) at that value, to give you the y-coordinate.

(d) Finding areas under graphs should have been covered in the chapter that taught you about the relationship between areas and integrals. Apply those techniques. ;)
 
Note: I corrected the typoed sign on 51x^2, which should be "plus" rather than "minus" (as in your graphic).

(c) To find the x-coordinate of the local min, set the derivative equal to zero, same as always:

. . . . .9x6+28x4+51x288x109x4+12x2+4=0\displaystyle \dfrac{9x^6\, +\, 28x^4\, +\, 51x^2\, -\, 88x\, -\, 10}{9x^4\, +\, 12x^2\, +\, 4}\, =\, 0

. . . . .9x6+28x4+51x288x10=0\displaystyle 9x^6\, +\, 28x^4\, +\, 51x^2\, -\, 88x\, -\, 10\, =\, 0

Presumably, you're expected to solve this within the specified software package. The solution (being an x-value slightly greater than 2) is the x-coordinate of the local min. I would expect that the software can evaluate f(x) at that value, to give you the y-coordinate.

(d) Finding areas under graphs should have been covered in the chapter that taught you about the relationship between areas and integrals. Apply those techniques. ;)
I tried to solve for x, but it's very difficult for me to get x:( . And for area under the graph, I used integration using points 0 and 2.2 but the result is negative *(-5.61) Is it correct?:confused::confused:
 
I tried to solve for x, but it's very difficult for me to get x.
I don't know your software, so I cannot tell you what commands to enter to get the solution. Sorry.

And for area under the graph, I used integration using points 0 and 2.2 but the result is negative *(-5.61) Is it correct?
Whether this answer is correct will depend upon your course. Are negative areas allowed, or are you supposed to use the absolute value of the function (or, in this case, the negative of the function) as the integrand (so that you get a positive area)? Check your class notes and textbook for examples. Whatever method they display there is what you should use here. ;)
 
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