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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+6x-2x^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+6x-2x^2 Since this is a parabola there should be 2 intercepts. Find the 2 x-intercepts 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 $f'(x)=0.$ Moreover, by finding $f''(x),$ you can classify the point; if $f'(x)=0\text{ and }f'#39;(x)<0$ then x is a maximum, and if $f'(x)=0\text{ and }f'#39;(x)>0$ then x is a minimum. If $f''(x)=0$ then you need to use some other method to classify the point. In your example, $y'(x)=6-4x$ and $y''(x)=-4;$ you can use these to verify what darksavior has already written.
March 9th, 2009, 09:49 PM   #4
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Re: I cannot figure this out at all?

Quote:
 Originally Posted by mattpi More generally, minimum or maximum values of a function f occur only at those values of x such that $f'(x)=0.$ Moreover, by finding $f''(x),$ you can classify the point; if $f'(x)=0\text{ and }f'#39;(x)<0$ then x is a maximum, and if $f'(x)=0\text{ and }f'#39;(x)>0$ then x is a minimum. If $f''(x)=0$ then you need to use some other method to classify the point. In your example, $y'(x)=6-4x$ and $y''(x)=-4;$ you can use these to verify what darksavior has already written.
Yes that is the most efficient way of doing it, but the problem is that this person probably taking algebra or geometry only. I dont think he knows anything about differientials yet.

March 10th, 2009, 05:41 AM   #5
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 Originally Posted by flyaway I have this problem i don't know how to do! Can anyone help? Let f(x)= 10+6x-2x^2 Determine whether f has a minimum or maximum value and find this value
If you're working within the context of algebra (rather than calculus), use what you know about graphing quadratics!

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 x-coordinate of the vertex using Vx=-B/2A and the y-coordinate via substitution.

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