I Is π^x - x^π < 0 Solvable by Graphing Techniques?

[bryan goh]

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

 

[jedishrfu]

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.

 

[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

 

[jedishrfu]

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.

 

[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

 

[Mark44]

Anyway i got this question from my math textbook where i study by myself.

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

 

[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 ∞.

 

[lavinia]

This is a standard Calculus 1 homework problem

 

[SlowThinker]

This is a standard Calculus 1 homework problem

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

 

[lavinia]

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

I never heard of the Lambert W function.

 

[SlowThinker]

I never heard of the Lambert W function.

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

 

[lavinia]

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

I don't know what this link tells you.

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

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

 

[SlowThinker]

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

So how do you solve that using Calculus 1 knowledge?

 

[Svein]

So how do you solve that using Calculus 1 knowledge?

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

 

[SlowThinker]

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

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

 

[bryan goh]

log pi/pi is constan right?

 

[NFuller]

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

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.

 

[bryan goh]

but what [x][0] must we take

 

[bryan goh]

x0 i mean

 

[NFuller]

but what [x][0] must we take

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 let's 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.

 

[Tanishq Nandan]

Can't we simply plot f(x)=log(x)/x,we know x>0...apply limits to find value of f(x) at 0,1 and infinity
It's clear that the derivative of the function will be positive till e and negative after that(indicating e is a point of maxima).
That's your graph done.
log(pi)/pi will be a straight line cutting the function at x=pi and some other point(we'll need a calculator to find that,i guess)

Your answer's that point till pi