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: 




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.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

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).`

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial