Selecting points at x = 0, π , π /2, π /4, the y-values, and the slopes, are simple combinations of the coefficients and √2/2.

Each coefficient leaves certain points and slopes of the curve unaffected.[br][br]How many times may the curve intersect the x-axis, between two red points?[br]For example, Consider adjacent points [math]{\small P = (x_p, y_p), \;\; Q=(x_p, y_p),\;\;\; x_q = x_p + π /4 }[/math] ( Q to the right of P). [br]How many times may the curve intersect the x-axis, between two adjacent points?[br][br]CASE: If [math]\;\;\; {\small y_p y_q < 0} [/math], then plainly f(x) = 0 somewhere between P and Q.[br]Can there be more than one intersection? If not, and I intersect:[br] [math]\;\;\;[/math] tangent to P,[br] [math]\;\;\;[/math] tangent to Q,[br] [math]\;\;\;[/math] or Segment PQ[br]with the x-axis ... which is the better approximation?[br][br]And other CASES?[br]E.g. Let R be the intersection of the tangents at P and Q. If [math] {\small x_p<x_r<x_q} [/math], triangle ΔPQR encloses the arc in some neighborhood of [math]{\small f(x_r)}[/math], but not necessarily the whole π/4 interval.[br][br]Can I draw a conservative envelope which is still a good approximation? I will need over- and under- estimates. Can I do this by forming quadrilaterals and/or triangles by some arrangement of [br] [math]\;\;\;[/math] 1. the given points (at intervals of π/4)[br] [math]\;\;\;[/math] 2. the vertical lines and tangents through them[br] [math]\;\;\;[/math] 3. intersections of 2?[br][br]Perhaps I can make a more informed estimate, by breaking up the function into its component parts...