if y= x^2+kx+ 3,determine the value(s) of k for which the minimum value of the function is an integer. Explain your reasoningadvanced function garde 11 please show me the steps

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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Since the equation of function is a quadratic equation and the leading coefficient is positive, hence, the vertex of parabola represents the minimum value of function such that:

`V(-b/(2a),(-Delta)/(4a))`

`a,b,c`  represent the coefficients of quadratic equation

`Delta = b^2 - 4ac`

`V(-k/2 , -(k^2 - 12)/4)`

Notice that the minimum value of the function is integer if -k is a multiple of 2, hence `k = 2n` .

Giving values to n yields:`k = -2 ; k = -4 ; k = -6..., k = -2n` .

Hence, evaluating the integer values of k yields `k in {-2n,.....,-6,-4,-2,0}.`

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quantatanu | Student, Undergraduate | (Level 1) Valedictorian

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In my answer the minimum of the function is 

 r = -k/2 

   = - 2*Sqrt[3-n]/2

   = -/+ Sqrt[3-n]


Therefore minimum values of the function are:

x^2 + k x + 3 for x = r, so

 

minimum of y = (3 - n) + (+/- 2 * Sqrt [3])*(-/+ Sqrt[3-n]) + 3

= (3 - n) - 2 * (3 - n)+3

= - (3-n)+3

= n

 

 
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quantatanu | Student, Undergraduate | (Level 1) Valedictorian

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The given equation is:

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

To get the minimum value we must get the minima by setting the first derivative = 0

y'(x) = 2 x + k 

suppose minimum point is x = r, so

y'(r) = 0

=> 2 r + k = 0

=> r = -k/2

Now we putr this minimum value in (1)

y = r^2 + k r + 3

  = (-k/2)^2 + k (-k/2) + 3

= k^2 / 4 - k^2/2 + 3

This minimum value of "y", we want to be an integer:

=> - k^2 / 4 + 3 = Integer = n (say)

=> k^2 / 4 = 3 - n

=> k^2 = 4 (3 - n)

=> k = +/- 2 * Sqrt[3-n], where n is an integer.

Now minimum value will be the minimum integer (positive integer I suppose) so n = 0 so

k = +/- 2* Sqrt[3 - 0]

   = +/- 2 * Sqrt [3]

 

 

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