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March 9th, 2009, 06:57 PM  #1 
Newbie Joined: Mar 2009 Posts: 1 Thanks: 0  I cannot figure this out at all?
I have this problem i don't know how to do! Can anyone help? Let f(x)= 10+6x2x^2 Determine whether f has a minimum or maximum value and find this value 
March 9th, 2009, 07:31 PM  #2 
Newbie Joined: Mar 2009 Posts: 3 Thanks: 0  Re: I cannot figure this out at all?
y= 10+6x2x^2 Since this is a parabola there should be 2 intercepts. Find the 2 xintercepts using quadratic formula. Then find the midpoint between the two intercepts. Plug it into the equation and you'll get y and that will be your minimum or maximum. The answer is. (1.5, 14.5) and it should be a maximum. 
March 9th, 2009, 07:50 PM  #3 
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  Re: I cannot figure this out at all?
More generally, minimum or maximum values of a function f occur only at those values of x such that Moreover, by finding you can classify the point; if then x is a maximum, and if then x is a minimum. If then you need to use some other method to classify the point. In your example, and you can use these to verify what darksavior has already written. 
March 9th, 2009, 09:49 PM  #4  
Newbie Joined: Mar 2009 Posts: 3 Thanks: 0  Re: I cannot figure this out at all? Quote:
 
March 10th, 2009, 05:41 AM  #5  
Senior Member Joined: Mar 2009 Posts: 318 Thanks: 0  Quote:
Since this is a quadratic function, then its graph is a parabola, so it has either a minimum or else a maximum, being of course the vertex. Since this is a negative quadratic (the coefficient on the leading term is 2, a negative number), the parabola opens downward, so the vertex is the maximum, rather than the minimum. Either complete the square to find the vertex, or else use the formula "h = b/(2a)".  
March 10th, 2009, 06:17 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 21,111 Thanks: 2326 
Completing the square: 10 + 6x  2x² = 14.5  2(x  1.5)², so it has a maximum value of 14.5.

March 10th, 2009, 09:42 PM  #7 
Member Joined: Mar 2009 From: San Bernardino, California Posts: 50 Thanks: 0  Re: I cannot figure this out at all?
By inspection you can see that it is a quadratic function and therefore is in graphical context, a parabola. It also has negative leading coefficient meaning that the parabola is concave down indicating the function has a maximum which should be it's vertex point as well. You can then find the xcoordinate of the vertex using Vx=B/2A and the ycoordinate via substitution.


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