+0  
 
0
83
4
avatar

For what constant k is 1 the minimum value of the quadratic 3x^2 - 15x + k over all real values of x? (x cannot be nonreal)

Guest Jul 2, 2017
edited by Guest  Jul 2, 2017

Best Answer 

 #1
avatar+4480 
+3

Let's say

y  =  3x2 - 15x + k                              Now let's get this into vertex form. Divide through by 3.

 

y/3  =  x2 - 5x + k/3                            Subtract  k/3  from both sides of this equation.

 

y/3 - k/3  =  x2 - 5x                             Add  (5/2)2  ,  or   25/4   to both sides of this equation.

 

y/3 - k/3 + 25/4  =  x2 - 5x + 25/4      Now we can factor the right side...

 

y/3 - k/3 + 25/4  =  (x - 5/2)2              Multiply through by  3  .

 

y - k + 75/4  =  3(x - 5/2)2                  Add  k  and subtract  75/4  to both sides.

 

y  =  3(x - 5/2)2 + k - 75/4

 

The vertex form of a parabola is:     y  =  a(x - h)2 + j     , where  (h, j)  is the vertex of the parabola.

(Normally you see k instead of j, but I used j since we already have a k in this problem.)

 

If the minimum value of  y  is  1, then the y-coordinate of the vertex of this parabola will = 1.

This means....

k - 74/4   =  1

 

k   =   1 + 75/4   =   79/4   =   19.75

 

_________________________________________________________________________________

_________________________________________________________________________________

 

 

Here is another approach...

 

y  =  3x2 - 15x + k             Take the derivative with respect to x on both sides.

 

dy/dx  =  6x - 15               The x value when dy/dx = 0 will be the x coordinate of the minimum,

                                          since this is a positive parabola. So, set it equal to 0 and solve for x.

0  =  6x - 15

 

x  =  15/6  =  5/2     Plug this value for x into the original equation to find the y coordinate of the min.

 

y  =  3(5/2)2 - 15(5/2) + k

 

y  =  -75/4 + k                   We are told that the y coordinate of the min. = 1,

                                          so plug in  1 for y and solve for k.

1  = -75/4 + k

 

k  =  1 + 75/4  =  19.75

hectictar  Jul 2, 2017
edited by hectictar  Jul 2, 2017
Sort: 

4+0 Answers

 #1
avatar+4480 
+3
Best Answer

Let's say

y  =  3x2 - 15x + k                              Now let's get this into vertex form. Divide through by 3.

 

y/3  =  x2 - 5x + k/3                            Subtract  k/3  from both sides of this equation.

 

y/3 - k/3  =  x2 - 5x                             Add  (5/2)2  ,  or   25/4   to both sides of this equation.

 

y/3 - k/3 + 25/4  =  x2 - 5x + 25/4      Now we can factor the right side...

 

y/3 - k/3 + 25/4  =  (x - 5/2)2              Multiply through by  3  .

 

y - k + 75/4  =  3(x - 5/2)2                  Add  k  and subtract  75/4  to both sides.

 

y  =  3(x - 5/2)2 + k - 75/4

 

The vertex form of a parabola is:     y  =  a(x - h)2 + j     , where  (h, j)  is the vertex of the parabola.

(Normally you see k instead of j, but I used j since we already have a k in this problem.)

 

If the minimum value of  y  is  1, then the y-coordinate of the vertex of this parabola will = 1.

This means....

k - 74/4   =  1

 

k   =   1 + 75/4   =   79/4   =   19.75

 

_________________________________________________________________________________

_________________________________________________________________________________

 

 

Here is another approach...

 

y  =  3x2 - 15x + k             Take the derivative with respect to x on both sides.

 

dy/dx  =  6x - 15               The x value when dy/dx = 0 will be the x coordinate of the minimum,

                                          since this is a positive parabola. So, set it equal to 0 and solve for x.

0  =  6x - 15

 

x  =  15/6  =  5/2     Plug this value for x into the original equation to find the y coordinate of the min.

 

y  =  3(5/2)2 - 15(5/2) + k

 

y  =  -75/4 + k                   We are told that the y coordinate of the min. = 1,

                                          so plug in  1 for y and solve for k.

1  = -75/4 + k

 

k  =  1 + 75/4  =  19.75

hectictar  Jul 2, 2017
edited by hectictar  Jul 2, 2017
 #2
avatar
+1

Thank you so much!

Guest Jul 2, 2017
 #3
avatar+76161 
+1

 

 

Very nice, hectictar....I probably would favor the Calculus approach, but the other is more comprehensible for the  Algebra student.....

 

 

 

cool cool cool

CPhill  Jul 2, 2017
 #4
avatar+4480 
+3

Thank you CPhill !!   smileysmiley   I'm glad you said that because I was thinking at first that it might have been unnecessary to put down the calculus method.

hectictar  Jul 2, 2017

9 Online Users

avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details