Find a positive number x such that the sum of 25x and 1/x is as small as possible

Question

Find a positive number  x  such that the sum of  25x  and  1/x  is as small as possible

mathmale · Answer

Are you taking a calculus course?

If so, let s(x) = sum of 25x and 1/x:        s(x)=25x+1/x.

What's the derivative of this?

anonymous · Answer

25x+(1/x)=y

Any negative number would work, since 25x+(1/x), as x is closer to -infinity goes to -infinity.  

Unless by "small", you mean closer to 0

anonymous · Answer

Calculus it is, I do not know how to do anything..
And by small, I assume close to 0, this is the question literally from my homework..

mathmale · Answer

"is as small as possible" is just fine.  Another way to pose this calculus problem is to say, "Minimize the sum of 25x and 1/x."

@laurenkp:  What is the derivative of 25x?  Use the power and constant coefficient rules.

Re-write 1/x as $$x ^{-1}$$    Then find the derivative of that, using the Power Rule.

anonymous · Answer

Derivative of 25x+1/x would be 25-1/x^2 but then after that I do not know what to do

anonymous · Answer

@mathmale

mathmale · Answer

Please equate your result to ZERO and solve for x.  Any solutions are called "critical values."

anonymous · Answer

Got it @mathmale thank you so much!!!!

mathmale · Answer

My pleasure!  Hope you'll post more problems here.  Hope to work with you again, too.