Horizontal tangent?

Question

Is there a value of c for f(x)=x^3+2x^2+cx for which there is exactly ONE horizontal tangent?

Answer 1

f'(x) = 3x^2 + 4x + c
To have exactly one horizontal tangent for f(x), f'(x) must have exactly one zero. Since it is a quadratic we know this will happen precisely when the discriminant (b^2 - 4ac for a quadratic ax^2 + bx + c) is zero, i.e.
4^2 - 4(3)(c) = 0
<=> 12c = 16
<=> c = 4/3.