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August 2nd, 2011, 04:41 PM  #1 
Newbie Joined: Aug 2011 Posts: 12 Thanks: 0  I need help with this
The profit function is given by P= 2X(square) + 72X198 Where X units is the quantity and P the profit. Find the maximum profit and the number of units where the maximum profit occurs. Sketch the curve of the profit function. 
August 2nd, 2011, 05:57 PM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: I need help with this
We are given: This is a downward opening parabola, and to find the point of maximum profit, we need to find the vertex of the parabola (h,k) by writing it in the form: Recall, the vertex of a parabola is where its global max/min occurs. Equating the two expressions, we find: Equating coefficients, we find: thus, we may write: ^2+450" /> Thus, maximum profit is 450 at a quantity of 18. To sketch the curve, begin by plotting the point (18,450). We may get 2 more points by finding the roots, or xintercepts. We do this by setting P(x) = 0. ^2+450=0" /> ^2225=0" /> ^2=225" /> Thus, one root is and the other is . So now plot the points (3,0) and (33,0) and connect the 3 points with a parabolic curve. Notes: We may also have found the vertex by completing the square: Or, we could use differential calculus: =18" /> =2(1^2+72(1198=450" /> To find the roots, we could have factored: thus we see the roots are x = 3, 33. Or we could have used the quadratic formula on : 
August 2nd, 2011, 07:06 PM  #3 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: I need help with this
Or... 
August 2nd, 2011, 07:11 PM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: I need help with this
You know, I actually thought about adding the axis of symmetry, but didn't. 
August 2nd, 2011, 07:29 PM  #5  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: I need help with this Quote:
 
August 2nd, 2011, 07:37 PM  #6 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: I need help with this
Although...you know...it was kind of implied in the differential calculus method, but one need only look at the quadratic formula to easily see where the axis of symmetry lies. There's nothing wrong with minimalism. :P 