For $x,y,z$ positive real numbers $3x+4y+7z=1$. Find minimum integer value of $1/x+1/y+1/z$
My solution like this: Using Cauchy-Schwarz inequality $$(3x+4y+7z)(1/x+1/y+1/z)\ge (\sqrt{3}+2+\sqrt{7})^2$$ $$(1/x+1/y+1/z)\ge 40,68$$ so min integer value of $1/x+1/y+1/z=41$
I wonder if there is more easy solution that not use Cauchy-Schwarz inequality. I seek more elementary solution. Thank you.
On
Alternative solution:
$3x+4y+7z=1 \implies z=\dfrac{1-3x-4y}{7}$
$\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{7}{1-3x-4y}=f(x,y)$
$\dfrac{\partial f}{\partial x}=-\dfrac{1}{x^2}+\dfrac{21}{(1-3x-4y)^2}, \quad \dfrac{\partial f}{\partial y}=-\dfrac{1}{y^2}+\dfrac{28}{(1-3x-4y)^2}$
$\dfrac{\partial f}{\partial x}=0 \iff 21x^2-(1-3x-4y)^2=0 \implies 21x^2= (1-3x-4y)^2 \qquad (1)$
$\dfrac{\partial f}{\partial y}=0 \iff 28y^2-(1-3x-4y)^2=0 \implies 28y^2= (1-3x-4y)^2 \qquad (2)$
Now, from $(1)$ and $(2)$ we can conclude this equality: $21x^2=28y^2 \iff 3x^2=4y^2$. Since we know that $x$ ans $y$ are positive real numbers, we know this equality will holds too: $y=\dfrac{\sqrt{3}}{2}x$.
We can plug this relation into $(1)$ and find:
$\sqrt{21}x=1-3x-4\dfrac{\sqrt{3}}{2}x \implies x=\dfrac{1}{3+2\sqrt{3}+\sqrt{21}}$.
Hence, we have:
$y=\dfrac{\sqrt{3}}{2(3+2\sqrt{3}+\sqrt{21})} \quad \text{and} \quad z=\dfrac{\sqrt{21}}{7(3+2\sqrt{3}+\sqrt{21})}$.
Finally, we know the minimum for $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}$ will be:
$3+2\sqrt{3}+\sqrt{21}+\dfrac{2(3+2\sqrt{3}+\sqrt{21})}{\sqrt{3}}+\dfrac{7(3+2\sqrt{3}+\sqrt{21})}{\sqrt{21}} \approx 40.676$.
Also, we can use AM-GM: $$(3x+4y+7z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=14+\frac{3x}{y}+\frac{4y}{x}+\frac{3x}{z}+\frac{7z}{x}+\frac{7z}{y}+\frac{4y}{z}\geq$$ $$\geq14+4\sqrt3+2\sqrt{21}+4\sqrt7.$$
The equality occurs for $\frac{3x}{y}=\frac{4y}{x}$, $\frac{3x}{z}=\frac{7z}{x}$ and $\frac{7z}{y}=\frac{4y}{z},$ which says that $14+4\sqrt3+2\sqrt{21}+4\sqrt7$ is a minimal value and since $$[14+4\sqrt3+2\sqrt{21}+4\sqrt7]+1=41,$$ we got an answer.