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 December 24th, 2017, 02:05 AM #1 Banned Camp   Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3 implicit function in inequality Can somebody, please, give me an example of problem that involve implicit function in inequality December 24th, 2017, 03:04 AM #2 Global Moderator   Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond ax + by < c with a, b, c real. Thanks from policer December 24th, 2017, 03:16 AM #3 Banned Camp   Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3 Nice. thanks!!! Thanks from greg1313 December 24th, 2017, 04:44 PM #4 Global Moderator   Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond Here's one with a twist! Minimize $\frac{2x^3+1}{4y(x-y)}$ subject to $x\ge-\frac12$ and $\frac xy>1$. I'll post my solution in a few days. December 27th, 2017, 09:56 PM   #5
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Quote:
 Originally Posted by greg1313 Here's one with a twist! Minimize $\frac{2x^3+1}{4y(x-y)}$ subject to $x\ge-\frac12$ and $\frac xy>1$. I'll post my solution in a few days.
$$\frac{2x^3+1}{4y(x-y)}=M$$

$$\frac{\partial M}{\partial y}=(2x^3+1)\frac{8y-4x}{(4y(x-y))^2}=0\implies2y=x$$

Note that $2x^3+1=0$ does not yield a solution as $2^{-1/3}\lt-\frac12$. Continuing,

$$\frac{2x^3+1}{4y(x-y)} \Leftrightarrow x=2y \Rightarrow \left(4y+\frac{1}{4y^2}\right)'=0 \Rightarrow 4-\frac{1}{2y^3}=0\Rightarrow y=\frac12,\quad x=1,\quad \min(M)=3$$

To verify,

$$\left(4y+\frac{1}{4y^2}\right)''=\frac{3}{2y^4}$$

which is always positive so $M$ is concave up and we have a minimum. Note that $M$ is not bounded from above.

I'll admit that a more rigorous analysis may be required to assert this is indeed the minimum of $M$. Can anyone provide one? Tags function, implicit, inequality Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post shaharhada Math 6 December 9th, 2017 03:56 AM Azkoul Calculus 1 August 31st, 2015 07:22 AM MATHEMATICIAN Abstract Algebra 5 February 12th, 2015 09:37 AM Robert Lownds Real Analysis 2 June 12th, 2013 12:27 AM OriaG Calculus 2 May 25th, 2013 02:56 PM

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