For what values of x is the tangent line to horizontal? If there are two such values and the value of y at one of those points is greater than 2 and at the other less than two, then the equation will have three distince real roots. Do you see why?
Find the equation at the tangent to at the point .
Hence deduce the values of k for which the equation will have 3 real and distinct roots
The equation of the tangent is
what do I do next?
Thank you
Well, do you know what the graph of a general cubic looks like? In general it looks like an "S" on its side- that is, the graph rises to some maximum value, then back down to a minimum value, then back up again (or the other way, first a minimum, then up to a maximu, then down again).
In either case, the maximum value and minimum values will be where the derivative is 0. If the maximum is above y= -2 and the minimum below it, the graph must cross y= -2 going up to the maximum, then cross it again going down to the minimum, and cross it a third time going up again.
In particular, has derivative which will be 0 at . At . is about -.224. That value of y will be greater than 2 for or or