MHB Can You Prove These Inequalities for Real Numbers a, b, and c?

  • Thread starter Thread starter anemone
  • Start date Start date
AI Thread Summary
The discussion centers around proving inequalities for real numbers a, b, and c, given the conditions a+b+c=6 and ab+bc+ca=9. Participants are tasked with demonstrating that 0<a<1<b<3<c<4. The thread also mentions an extension of the deadline for the previous problem of the week (POTW) to encourage more submissions. The goal is to engage members in solving mathematical inequalities effectively. Overall, the focus remains on the proof of the specified inequalities under the given constraints.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Here is this week's POTW:

-----

Let $a<b<c$ be real numbers such that $a+b+c=6$ and $ab+bc+ca=9$. Prove that $0<a<1<b<3<c<4$.

-----

 
Physics news on Phys.org
Hi all!

Just to let you all know that I will extend the deadline to solve last week's POTW (which actually due today) to next week, with the great hope to receive any submission of solution from the members! (Happy)
 
Let $abc=k$, this makes $a,\,b$ and $c$ become the roots of the cubic equation $x^3-6x^2+9x-k=0$.

The cubic equation can be rewritten as $x(x-3)^2=k$.

Plot the graph of $f(x)=x(x-3)^2$ and the horizontal line of $y=k$ with condition $0<k<4$ on the same diagram. Since the roots are real, the line $y=k$ can only be moved in such a way that it always cuts the curve at 3 distinct points in such a way that $0<a<1<b<3<c<4$.

[TIKZ][scale=3]
\draw[help lines] (0,0) grid (5,4);
\draw[thick,->] (0,0) -- (4.5,0) node[anchor=north west] {x axis};
\draw[thick,->] (0,0) -- (0,4.5) node[anchor=south east] {y axis};
\draw[ domain=-0.2:4, samples=100] plot (\x,\x^3 - 6*\x^2 +9*\x);
\draw (0,3.2) -- (4,3.2);
\foreach \x in {0,1,2,3,4}
\draw (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\x$};
\foreach \y in {0,1,2,3,4}
\draw (1pt,\y cm) -- (-1pt,\y cm) node[anchor=east] {$\y$};
\node at (3.6,2) {\large $y=f(x)$};
\node at (4.2,3.2) {\large $y=k$};
\draw[ densely dashed,color=blue] (0,4) -- (4,4);
\draw[ densely dashed,color=blue] (4,0) -- (4,4);
\draw[ densely dashed,color=blue] (0.52,0) -- (0.52,3.2);
\draw[ densely dashed,color=blue] (1,0) -- (1,3.2);
\draw[ densely dashed,color=blue] (1.58,0) -- (1.58,3.2);
\draw[ densely dashed,color=blue] (3.9,0) -- (3.9,3.2);
\node at (0.52,-0.08) {$a$};
\node at (1.58,-0.08) {$b$};
\node at (3.9,-0.08) {$c$};
[/TIKZ]
 
Back
Top