What are the minimum and maximum values?

For what values of c does the curve have maximum and minimum points for the given function?

• For what values of c does the curve have maximum and minimum points for the given function f(x) = 6x^3 + cx^2 +6x Answer choices = +/- 6 sqrt(3) = +/- sqrt(103) = +/- 7 sqrt ...show more

• Answer:

  df/dx = 18x^2+2cx+6. Local maxima/minima occur where 18x^2+2cx+6 = 0 3x^2+2bx+1 = 0 where b=c/6. This quadratic has real roots if "b^2>=4ac" ie if (2b)^2 >= 4*3*1 b^2 >= 3 b <=-√3 or b >= +√3. So maxima/minima exist when c <=-6√3 or c >= +6√3....(*) since c=6b. 6√3 = √(36x3) = √108. 7√2 = √(49x2) = √98. So, of the options available, only the first (±6√3) meets the criterion (*).