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Solve the equation $2^x + 5^x = 3^x + 4^x$. I can figure out two special solutions $x=0$ and $x=1$, and I try to prove that they are the only two solutions. However, I find it hard to do so because I can't prove the monotony given there are also exponential in the derivative. Any hints to that?

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2 Answers 2

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$$2^x+5^x=3^x+4^x \iff5^x-4^x=3^x-2^x$$ Lagrange's theorem applies (intermediate value theorem) functions: $$f:[4, 5]\to \mathbb R,\ g:[2, 3]\to \mathbb R,\ f(u)= u^x,\ g(v)=v^x$$ Exist $c\in [4, 5]$ and exist $ d\in [2, 3] $ so $5^x-4^x = xc^{x-1}$ and $3^x-2^x=xd^{x-1}$.

The equation is written as equivalent: $$xc^{x-1}= xd^{x-1}.$$ It follows that equation solutions are $0$ and $1.$

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Since $f(a)=a^x$ is concave for $x\in[0,1]$ and convex for $x\not\in(0,1)$, the definitions of concavity and convexity say $$ \begin{align} \color{#C00}{3^x}+\color{#090}{4^x} &=\color{#C00}{\left(\frac23\cdot2+\frac13\cdot5\right)^x}+\color{#090}{\left(\frac13\cdot2+\frac23\cdot5\right)^x}\tag1\\[3pt] &{\ge\atop\le}\color{#C00}{\left(\frac23\cdot2^x+\frac13\cdot5^x\right)}+\color{#090}{\left(\frac13\cdot2^x+\frac23\cdot5^x\right)}{{\quad\text{if }x\in[0,1]}\atop{\quad\text{if }x\not\in(0,1)}}\tag2\\[6pt] &=2^x+5^x\tag3 \end{align} $$ Explanation:
$(1)$: write $3$ and $4$ as convex combinations of $2$ and $5$
$(2)$: definition of $\text{concavity}\atop\text{convexity}$
$(3)$: combine like terms

Furthermore, equality holds only if $f$ is linear on $\{2,3,4,5\}$, and that happens iff $x\in\{0,1\}$.

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