No. of real solutions in exponential equation

[jacks]

no. of real solution of the equation $3^x+4^x+5^x = x^2$

 

[mathbalarka]

Re: no. of real solution in exponential equation.

The RHS grows superexponentially whether the LHS is quadratic. So, for obvious reasons, they intersects each other at most finitely often. A more close inspection of the behaviour would lead to the fact that they do indeed intersect ones.

PS I have no formal proof of this piece of problem, unfortunately.

Balarka
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[Opalg]

Re: no. of real solution in exponential equation.

The RHS grows superexponentially whether the LHS is quadratic. So, for obvious reasons, they intersects each other at most finitely often. A more close inspection of the behaviour would lead to the fact that they do indeed intersect ones.

PS I have no formal proof of this piece of problem, unfortunately.

That argument looks correct to me. As $x$ goes from $-\infty $ to $0$, the LHS increases from $0$ to $12$, and the RHS decreases from $\infty$ to $0$. So the graphs must cross exactly once. For $x>0$ the LHS increases (very fast!) from $12$ to $\infty$ and the RHS increases much more slowly. If you differentiate both functions I am sure you will find that the LHS increases faster than the RHS for all positive $x$.

 

[eddybob123]

Re: No. of real solution in exponential equation

Hint:

Spoiler

Considering the function $y=3^x+4^x+5^x-x^2$, use the Intermediate Value Theorem to show that there is at least one solution, then use Rolle's Theorem to show that it is unique.

 

[mathbalarka]

Re: No. of real solution in exponential equation

I am sure you will find that the LHS increases faster than the RHS for all positive x

That is evident, as I mentioned before, since LHS grows superexponentially, i.e., \(\displaystyle << 5^x\) whereas the RHS is quadratic. I think this is sufficient to prove the fact.

PS I see eddybob just posted a rigourus argument of the fact here