# I Π^x and x^π

1. Jun 26, 2017

### bryan goh

Guys, could you help me how to solve the inequality π^x - x^π < 0??

2. Jun 26, 2017

### Staff: Mentor

The easiest way to solve this is to use the desmos graphing calculator site:

https://www.desmos.com/calculator

and type in: pi^x - x^pi

It will show you a plot of the curve from which you can see where the zeros are and where the <0 segment is.

3. Jun 26, 2017

### bryan goh

but if we're not allowed to use any calculator?? because my school doesn't allow us to use calculator for most of my math lesson

4. Jun 26, 2017

### Staff: Mentor

Okay, but since you've posted it, you could look at the graph and then see if you can devise a strategy to solve it.

One obvious solution is: $\pi^\pi - \pi^\pi$ which is one of its zeros.

Next, what math course is this for?

Can you use an approximation strategy like evaluating a few terms in its Taylor series?

Also you can try x=0, x=1... and attempt to plot it.

5. Jun 26, 2017

### bryan goh

yeah, at first i think the solution is x<π. But when i look at the graph, there are another solution that make the inequalities become smaller than zero. Anyway i got this question from my math textbook where i study by myself. and yes i can use a bit of approximation of taylor series

6. Jun 26, 2017

### Staff: Mentor

Please post textbook problems in the Homework & Coursework sections, not here in the technical math sections.

7. Jun 26, 2017

### olgerm

Function $f(x)=π^x - x^π$ is continuous. Find values of x when f(x)=0 aka $π^x - x^π=0$. Ranges where f(x)<0 aka $π^x - x^π<0$ must be between those x values, in range between -∞ and smallest such x value or in range between biggest such x value and ∞.

8. Jun 28, 2017

### lavinia

This is a standard Calculus 1 homework problem

9. Jun 28, 2017

### SlowThinker

Really? I haven't heard of Lambert W function until well out of university. But then it wasn't a mathematical university.

10. Jun 29, 2017

### lavinia

I never heard of the Lambert W function.

Last edited: Jun 29, 2017
11. Jun 29, 2017

### SlowThinker

Wolframalpha gives the solution in terms of the LambertW function. Is there an easier expression for the 2.3821790879930187746?

12. Jun 29, 2017

### lavinia

I don't know what this link tells you.

I think you want to solve $log(x)/x > log(π)/π$ since

$π^{x} - x^{π} <0 ⇒ e^{xlog(π)} < e^{πlog(x)} ⇒ xlog{π} < πlog{x}$

13. Jun 29, 2017

### SlowThinker

So how do you solve that using Calculus 1 knowledge?

14. Aug 10, 2017

### Svein

Start with observing that $\frac{\log(\pi)}{\pi}$ is a constant.

15. Aug 13, 2017

### SlowThinker

And continue how? Remember this is not a proof of existence, we're looking for the value of x where $\log x/x=\log\pi/\pi$.

16. Sep 2, 2017

### bryan goh

log pi/pi is constan right?

17. Sep 2, 2017

### NFuller

If you can approximate this and this is question is from a calculus book, then this sounds like a problem were you should use Newton's method for finding the roots of a function.

18. Sep 2, 2017

### bryan goh

but what [x][0] must we take

19. Sep 2, 2017

### bryan goh

x0 i mean

20. Sep 2, 2017

### NFuller

You make a guess of $x_{0}$ which you think is close to the solution. We know $\pi$ is one solution of $\pi^{x}-x^{\pi}=0$ so lets see if there is another solution smaller than $\pi$. Try using $x_{0}=0$ for simplicity and you should get the other solution.

Edit: Sorry, looking at the graph you should probably pick $x_{0}=2$. The problem with Newton's method is that if you pick a value of $x_{0}$ and there is a hill or valley between that $x_{0}$ and the solution, the method does not converge.

Last edited: Sep 2, 2017