Weird maximum and minimums question?

For any constant c, define the function Fc with the formula Fc(x)=x^3+2x^2+cx





Graph y=Fc(x) for these values of the parameter c: c=-1,0,1,2,3,4.





a.For what values of the parameter c will Fc have no local maximum or minimum? Use calculus





b.Are there any values of the parameter c for which Fc will have exactly one horizontal tangent line?





Ok I graphed it, but how do you get the local maximum and minimum by using calculus? and how do you get the values for the horizontal tangent line? I'd appreciate any help.





Thank You.

Weird maximum and minimums question?
yes, very!
Reply:Fc(x) = x^3+2x^2+cx


Fc'(x) = 3x^2+4x+c





a. You are looking for the case where Fc'(x) = 0 has no solutions. Fc'(x) is a quadratic. So check the discriminant.





4^2-4*3*c %26lt; 0


16-12c %26lt;0


16 %26lt; 12c


4/3 %26lt; c





b. You're looking for the case where there's exactly 1 solution for Fc'(x) = 0. Once again, check the discriminant.





4^2-4*3*c = 0


16-12c = 0


16 = 12c


4/3 = c
Reply:Differentiate fc(x) and equate to zero to get the stationary points - substitute into Fc(x) to get the y-coordinates.





Differentiate again and stick your values for x in. When your answer is positive, the point is a local minimum. When your answer is negative, the point is a local maximum.
Reply:Take the derivative of Fc(x) with respect to x





you get





dFc/dx = 3x/3 + x + c





for there to be no local min or max dFc/dx cannot = 0





so find the ranges where 3x/3+x+c = 0, for each of those values of c (plug in c, solve for x), then say that the range was every number but those values you get for x.