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 Algebra Pre-Algebra and Basic Algebra Math Forum

June 7th, 2015, 06:43 PM   #1
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find minimum with log

Hi, I need help with this two problems. Please someone show me the way to solve them! Thanx
Attached Images 11119227_10153410771519700_48842836_n.jpg (21.1 KB, 11 views) 11420081_10153410771564700_2080206467_n.jpg (17.7 KB, 8 views) June 7th, 2015, 08:37 PM #2 Senior Member   Joined: May 2015 From: Varanasi Posts: 110 Thanks: 5 Math Focus: Calculus You should use derivatives to find the minimum value, i think this question is of calculus, differentiate the function with respect to x and then solve it for the point where it is becoming negative.Find y in terms of x, then find dy/dx and then solve the inequality dy/dx<0, find x and then find the value of y at this x. June 7th, 2015, 10:09 PM #3 Math Team   Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244 I think I have a solution for the second one without calculus. From the restriction that $xy = 8$, we may obtain an upper bound on $x$ and $y$. That is, $2\leq x\leq 16$ and $\dfrac{1}{2}\leq y\leq 4$. Taking the log base 2 of both sides of the equation $xy = 8$, we obtain $$\log_2xy = \log_2x + \log_2y = 3$$ So that $$\log_2y = 3 - \log_2x$$ From this, we obtain \begin{align*} &\,\,\,\,\,\,\,\,\log_2\,x\cdot\log_2\,y \\\\ &= \log_2x(3 - \log_2x) \\\\ &= 3\log_2x - (\log_2x)^2 \\\\ &= -\left((\log_2x)^2 - 3\log_2\,x + \dfrac{9}{4}\right) + \dfrac{9}{4} \\\\ &= \dfrac{9}{4} - \left(\log_2x - \dfrac{3}{2}\right)^2 \end{align*} This expression is at its maximum when the second term is $0$. This occurs when $\log_2x = \dfrac{3}{2}$. This implies that $x = 2^{2/3}$, which is within the permitted range. Thus the maximum is $\dfrac{9}{4}$. The minimum occurs when the second term is as large as possible. We try the extreme values of $x$, namely $2$ and $16$. If $x = 2$, $\log_2x\cdot\log_2y = 2$ If $x = 16$, $\log_2x\cdot\log_2y = -4$ Of these, $-4$ is the smallest. Thus the minimum is $-4$. Thanks from matisolla Tags find, log, minimum 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 thehurtlooker Algebra 6 April 11th, 2013 06:17 AM Albert.Teng Algebra 3 October 16th, 2012 06:34 PM hgphtgi Linear Algebra 13 September 1st, 2012 04:30 PM sivela Calculus 2 January 21st, 2011 02:19 PM marshell Number Theory 6 December 16th, 2006 11:49 PM

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