Finding max/min values of challenging function.

[the_dialogue]

Homework Statement

I am given two functions:
$$f(x)=\frac{x^{25}}{1.0001^x}$$

$$g(x)=\frac{1.0001^x}{1.0001^x+x^{25}}$$

I'm first asked to find the values of x for which f(x) reaches a maximum and g(x) reaches the minimum.

Secondly, I'm asked to find the actual max value of f(x) and min value of g(x) to 5 significant digits.

The Attempt at a Solution

I have completed the first part and found that f(x) reaches it's max value and g(x) reaches it's minimum value at the same 'x'. This x value is 25/ln(1.0001).

However, I do not know how to do the second part of the question. Obviously, the x value is incredibly large, since ln(1.0001), it's denominator, is very small. This makes simple calculator use impossible. How then do I calculate the value of f(x) and g(x) at this x?

I have considered of transforming f(x) into a taylor series by expanding x^25. Maybe then I can approximate the function by its taylor series expansion for 5 digits. Is this correct? Furthermore, can I have a hint for evaluating g(x) at its minimum point?


Thank you,
Alex.

 

[drpizza]

I just jotted a little bit on a piece of paper; I tried this for f(x): first take the natural log of that function, then later, e^(what it simplifies to). I didn't follow it through to completion, but a couple ln(1.0001)'s canceled out. Later, I ended up with (25/e)^25 * 1/ln(1.0001) (hopefully I don't have a careless error)

Looking at the reciprocal of the ln there, its value is 10000.499991667...
And, the quantity raised to the 25th power isn't so large (written that way) that it makes a calculator useless.

 

[benorin]

Do you see a way to write g(x) in terms of f(x)?

Also, the critical point is x=25/ln(1.0001) and f(25/ln(1.0001))=1.23504x10^124 and g(25/ln(1.0001))=8.0969x10^-125.

I used www.quickmath.com[/url] for the calculations, ( the [PLAIN]http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=algebra&s2=simplify&s3=basic will do calculations for you)

 

[the_dialogue]

Yes, I expressed g(x) in terms of f(x). But where does this get me?

I wasn't sure how you managed to calculate those values either. Could it be done with a hand-calculator or does it involve other tools (like quickmath.com)?