# Roots of equation.

1. Oct 10, 2009

### icystrike

1. The problem statement, all variables and given/known data
Show that there is only one stationary point of the curve $$y=e^{x/2} - ln (x)$$, where x>0 and determine the nature of the stationary point.

My approach:

dy/dx = $$0.5e^{x/2} - 1/x$$

When dy/dx=0 For stationary point.

Thus, through algebraic manipulation,

$$ln(2)-0.5x=ln(x)$$

Since,$$ln(2)-0.5x$$ is a deacreasing function ,

and $$ln(x)$$ is a increasing function with an horizontal asymptote of x=0.

Therefore , there is only an intersection at coordinate (1.13429 , 0.126007) by newton-raphson method.

Correct me if I am wrong or please show me a prove that is more elegant.

Last edited: Oct 10, 2009
2. Oct 10, 2009

### CFDFEAGURU

You are trying to find a zero slope (where the derivative vanishes) not a root. I haven't worked this out yet myself, but your methodology looks correct to me.

Thanks
Matt

3. Oct 10, 2009

### icystrike

A million thanks (=
Have a great day!

4. Oct 10, 2009

### CFDFEAGURU

Remember you can always graph your original function before you perform any differentiation to see where there might be a slope of zero. This can be very helpful if you have a difficult function to diferentiate.

Thanks
Matt

5. Oct 10, 2009

### icystrike

Noted (=

6. Oct 11, 2009

### Mentallic

Oh wow... this looks quite elegant, if it's legitimate
I am completely lost on what you've done, and after seeing that for this equation, solving for x when dy/dx=0 cannot be done as simply as through my usual routine, I'm curious to what you've done.
If you don't mind, could you please elaborate on what the "through algebraic manipulation" actually involved and thus how you resulted in the next line?