To find the location of roots of an equationProve that eπx−π+πex−e+ee+ππx−e−π=0. has one real root in (e,π) and other (π,π+e).

Expert answers to "To find the location of roots of an..."

Trent Fuller

2022-04-10

To find the location of roots of an equation
Prove that

\frac{e^{π}}{x - π} + \frac{π^{e}}{x - e} + \frac{e^{e} + π^{π}}{x - e - π} = 0

. has one real root in

(e, π)

and other

(π, π + e)

dabCrupedeedaejrg

Beginner2022-04-11Added 10 answers

Let

f (x) = \frac{e^{π}}{x - π} + \frac{π^{e}}{x - e} + \frac{e^{e} + π^{π}}{x - e - π} .

Thus,

lim_{x \to e^{+}} f (x) = + \infty

and

lim_{x \to π^{-}} f (x) = - \infty .

But f is continuous on

(e, π)

, which says that there is a root of the equation on (e,π).
By the same way we can obtain that there is a root on

(π, e + π)

.
But,

f (x) = 0

is a quadratic equation...
Another way.
Rewrite our equation in the form

g (x) = 0

, where

g (x) = e^{π} (x - e) (x - e - π) + π^{e} (x - π) (x - e - π) + (e^{π} + π^{π}) (x - e) (x - π) .

Now, check g(e),

g (π)

and

g (e + π)

.
Easy to see that

g (e) > 0, g (π) < 0

and

g (e + π) > 0

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