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How do I find the maximum value of the function f(x)=5^x-x^5 on the interval 0<=x<=2? I know that to find the maximum value I need to find the derivative of the function and then set it to zero. And i got:  f'(x)=5^x(ln5)-5x^4 0=5^x(ln5)-5x^4 x^4=5^(x-1)(ln5) And this is where i got stuck, i cannot solve the above equation. And i know there is an answer since using guess and check i was about to come close to the answer. I asked my teacher about this first, according to him, it's unsolveable using algebra. and i do not need to know about it at grade 12 high school(but i still wish to know). So basically i'm asking if you can show me how to solve this, or tell me what i would need to know to be able to solve this.

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Luca B. eNotes educator | Certified Educator

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You have followed the correct steps when you have differentiated the function and then you have tried to solve `f'(x)=0.`

Notice that the equation you need to solve is transcedental, hence you should use numerical or graphical methods to find the roots.

You should consider two functions such that:

`g(x)=x^4 and h(x) = 5^(x-1)*ln 5`

You need to sketch the graphs of these functions and the roots of the equation g(x)=h(x) represent the x coordinates of the points of intersection of graphs.

Notice that x coordinate of point of intersection between the red and black curves is in `(1,2), ` hence the function reaches its extreme point at a value of `x in (1,2).`

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etotheeyepi | Student

Numerical methods you could use to solve this question and you could look further into it by yourself.

You might try Maclaurin Series.

I have not tried it, so I don't know for sure, but I bet graphing the function with Excel is faster.

swain-ashe | Student
To faiz3507 Thanks for your effort, but I think you misread the question since the answer are not restricted to integers only. To etotheeyepi thanks for trying to solve the question, I actually got a similar answer, by guess and checking (when i first got the question, it was multiple choice so i was able to get it by subbing in the choices given). But it appears that only way I can solve it is either by graphing or just getting an estimate value only. To sciencesolve thank you, I realized that graphing could be a way to solve this question. It's just that if this question appeared on a test I would have no way to accurately draw the graphs and get an exact answer, so I was wondering if you could just tell me what type of numerical methods there are to solve this question and i could look further into it by myself.