Solve Root of Equation Q1a: Help Here

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Discussion Overview

The discussion revolves around solving a mathematical problem related to finding the critical points of a function defined by the equation $$c(x)=10-20(e^{-\frac{x}{5}}-e^{-\frac{3x}{4}})$$. Participants explore the process of deriving the first and second derivatives to analyze the function's behavior, specifically focusing on identifying minimum points. The context includes concerns about whether the question is part of a graded assignment or exam.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents the function and derives its first derivative, $$c'(x)$$, and sets it to zero to find critical points.
  • The same participant provides a detailed solution leading to the critical point $$x = \frac{20}{11}\ln \frac{15}{4}$$ and suggests that the second derivative should be evaluated to confirm a minimum.
  • Several participants question whether the problem is part of a graded assignment, indicating a concern about academic integrity and forum policy.
  • One participant expresses confusion regarding the nature of the question, suggesting it may be metaphorical, while others clarify that they are asking about its literal context.
  • A participant mentions that if the problem is from a practice test, assistance could be provided without issue.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the question, with some believing it to be metaphorical while others assert it is a literal inquiry. There is no consensus on whether the problem is part of a graded assignment, leading to ongoing discussion about the appropriateness of providing help.

Contextual Notes

There are unresolved questions regarding the classification of the problem as part of a graded assignment or a practice test, which affects the willingness of participants to engage in providing assistance.

wonguyen1995
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View attachment 4507

Can you help me question 1a?
 

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$$c(x)=10-20(e^{-\frac{x}{5}}-e^{-\frac{3x}{4}})$$

To find the critical points we have to find the points $x$ such that $c'(x)=0$:

$$c'(x)=20\left ( e^{-\frac{x}{5}}\left ( -\frac{1}{5} \right )-e^{-\frac{3x}{4}}\left ( -\frac{3}{4} \right ) \right )=-4e^{-\frac{x}{5}}+15e^{-\frac{3x}{4}} $$
$$c'(x)=0 \Rightarrow -4e^{-\frac{x}{5}}+15e^{-\frac{3x}{4}}=0 \Rightarrow 4e^{-\frac{x}{5}}=15e^{-\frac{3x}{4}} \\ \Rightarrow \ln \left ( 4e^{-\frac{x}{5}} \right ) =\ln \left ( 15e^{-\frac{3x}{4}} \right ) \Rightarrow \ln 4+ \ln e^{-\frac{x}{5}} =\ln 15+\ln e^{-\frac{3x}{4}} \\ \Rightarrow \ln 4 -\frac{x}{5} =\ln 15-\frac{3x}{4} \Rightarrow \frac{3x}{4}-\frac{x}{5} =\ln 15-\ln 4 \\ \Rightarrow \frac{15x}{20}-\frac{4x}{20} =\ln \frac{15}{4} \Rightarrow \frac{11x}{20} =\ln \frac{15}{4} \\ \Rightarrow x =\frac{20}{11}\ln \frac{15}{4}$$

To show that at this point the function achieves its minimum we have to calculate the second derivative of the function at this point, $c''\left ( \frac{20}{11}\ln \frac{15}{4}\right )$, and it should be $>0$.
 
Is this question part of an exam for credit?
 
Ackbach said:
Is this question part of an exam for credit?

what do you mean?
 
wonguyen1995 said:
what do you mean?

Hi wonguyen1995,

What Ackbach meant was whether the question you posted is a question that you will get marks for. Whether it's part of a graded assignment or an exam given in school?
 
Sudharaka said:
Hi wonguyen1995,

What Ackbach meant was whether the question you posted is a question that you will get marks for. Whether it's part of a graded assignment or an exam given in school?
i know that but i think this question is metaphor
Well forget it
 
wonguyen1995 said:
i know that but i think this question is metaphor
Well forget it

I am not quite sure what you mean. My question was a literal one, not metaphorical. Perhaps your earlier question was a metaphor?

In any case, I'm asking my question because it is against MHB policy knowingly to help with graded problems. Now, if the page you posted is from a practice test, there's no issue at all. But if it's from a graded test, then we will need to close this thread.
 
Ackbach said:
I am not quite sure what you mean. My question was a literal one, not metaphorical. Perhaps your earlier question was a metaphor?

In any case, I'm asking my question because it is against MHB policy knowingly to help with graded problems. Now, if the page you posted is from a practice test, there's no issue at all. But if it's from a graded test, then we will need to close this thread.

Sorry. Maybe i think a lot. it is the practice test in the previous semester. So can you help me figure out one ?
 

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