This is from Apostol's Calculus, Vol. II, Section 9.15 #11:
Find the maximum of $f(x,y,z)=\log x + \log y + 3 \log z$ on that portion of the sphere $x^2+y^2+z^2=5r^2$ where $x>0,\,y>0,\,z>0$. Use the result to prove that for real positive numbers $a,b,c$ we have $abc^3\le 27\left(\frac{a+b+c} 5\right)^5$
I had no trouble with the first part, using Lagrange's Multipliers. The maximum of $f$ subject to this constraint is $f(r,r,\sqrt 3 r )=5\log r + 3\log \sqrt 3$, and this answer matches the book's.
Now I see how we can take $f(a,b,c)=\log(abc^3)$. Then define $r>0$ by $a^2+b^2+c^2=5r^2\implies r=\sqrt {\frac{a^2+b^2+c^2} 5}$, so we can conclude that
$abc^3\le3^{3/2}\left(\frac{a^2+b^2+c^2}5\right)^{5/2}$
But this is a looser bound (for some numbers) than the one suggested by the text. In particular, if we consider $a=\frac 1 4,b= 1, c=1$, then $27\left(\frac{a+b+c} 5\right)^5<3^{3/2}\left(\frac{a^2+b^2+c^2}5\right)^{5/2}$ so I have the feeling that "I can't get there from here", at least not using the method suggested. Am I correct?
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