SOLUTION: Find k so that the minimum value of f(x) = x^2 + kx + 8 is equal to the maximum value of g(x) = 1 + 4x -2x^2

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Question 1189255: Find k so that the minimum value of f(x) = x^2 + kx + 8 is equal to the maximum value of g(x) = 1 + 4x -2x^2
Found 2 solutions by ikleyn, Solver92311:
Answer by ikleyn(52793)   (Show Source): You can put this solution on YOUR website!
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Find k so that the minimum value of f(x) = x^2 + kx + 8 is equal to the maximum value of g(x) = 1 + 4x -2x^2
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                    Solve it in three steps. 


 
STEP 1.   Find the maximum value of g(x) = 1 + 4x - 2x^2 .



    Find   = " -b/(2a) " =  =  = 1.


    Then  = 1 + 4*1 - 2*1^2 = 1 + 4 - 2 = 3.



STEP 2.   Find the minimum value of f(x) = x^2 + kx +8 .



      =  -  + 8.


    The minimum value of f(x) is   = -  + 



STEP 3.   Find k .


    We will find the value of k from this equation 

        = ,

    which is

       3 = -  + .


    Simplify and find k

       12 =  + 32

        = 32 - 12

        = 20

       k = +/- .


ANSWER.  There are two values for k:   =   and  - = -.

Solved.



Answer by Solver92311(821)   (Show Source): You can put this solution on YOUR website!






is a quadratic polynomial with a positive lead coefficient so it has a minimum value where






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is a quadratic polynomial with a negative lead coefficient so it has a maximum value where







So the problem is to solve:



for

You can do the necessary arithmetic.

John

My calculator said it, I believe it, that settles it

From
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