Homework Help

the parabola y=3x^2+18x+19 will have a minimum at what?  

user profile pic

canucksalltheway | Student, Grade 10 | eNoter

Posted September 28, 2010 at 11:37 AM via web

dislike 1 like

the parabola y=3x^2+18x+19 will have a minimum at what?

 

Tagged with math

5 Answers | Add Yours

user profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted September 28, 2010 at 11:48 AM (Answer #1)

dislike 1 like

y= 3x^2 + 18x + 19

To find the minimum, first we need to determine the derivative's zeros, then substitute in y:

y' = 6x + 18 = 0

==> 6x = -18

==> x= -18/6 = -3

Then the function y has a minimum value at x= -3

==> The minimum value is:

y= 3(-3)^2 + 18(-3) + 19

   = 27 - 54 + 19 = -8

Then the parabola has a minimum value at the point (-3, -8)

or f(-3) = -8

user profile pic

neela | High School Teacher | Valedictorian

Posted September 28, 2010 at 12:01 PM (Answer #2)

dislike 1 like

y = 3x^2+18x+19

To find the minimum value of y.

y = 3x^2+18x+19

y/3 = (3x^2+18x+19)/3

y/3 = x^2+6x+ 19/3.

y/3 = x^2+6x+9 - 8/3

y/3 = (x+3)^2 - 8/3.

y/3 > -8/3, as (x+3) is always greater than zero.

y/3 has the minimum  value -8/3 when (x+3)^2 is minimum or x+3 = 0 . Or when x =-3.

y is minmum value (-8/3)*3 = -8 , when x = -3.

user profile pic

mikedbz | Student , Undergraduate | eNotes Newbie

Posted September 28, 2010 at 11:50 AM (Answer #3)

dislike 0 like

y=3(x+6x)+19

=3(x+6x+9)+10-9

=3(x+3)^2 -17

 

It will have a minimum of -17 when  x = -3.

(somebody correct me if I'm wrong!)

user profile pic

phucnguyen08 | Student , College Freshman | Honors

Posted September 28, 2010 at 12:31 PM (Answer #4)

dislike 0 like

y=3(x^2+6x+9)+1

y=3(x+3)^2+1

We have 3(x+3)^2>=0

Therefore y=3(x+3)^2+1>= 1

so y minimum=1 when x=-3

user profile pic

william1941 | College Teacher | Valedictorian

Posted September 28, 2010 at 4:44 PM (Answer #5)

dislike 0 like

We have the function f(x) = 3x^2 + 18x + 19

Differentiating it f'(x) = 6x + 18

Now the extreme of the function is at the point f'(x) = 0

=>6x + 18 = 0

=> x = -18 / 6

=> x= -3

Also, f''(x) = 6 which is positive for all x. Therefore at x = -3 the function has a minimum.

The minimum value is:

3* (-3)^2 + 18*(-3) + 19 = 3*9 - 54 + 19 = - 8

Therefore the minimum value is -8 at x = -3

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes