Solve Root of Equation Q1a: Help Here

[wonguyen1995]

View attachment 4507

Can you help me question 1a?

 

[mathmari]

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

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 $>0$.

 

[Ackbach]

Is this question part of an exam for credit?

 

[wonguyen1995]

Is this question part of an exam for credit?

what do you mean?

 

[Sudharaka]

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?

 

[wonguyen1995]

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

 

[Ackbach]

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.

 

[wonguyen1995]

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 ?