Over the Interval 0 ≤x ≤2pi, how many points on the graph of the function y=cosx have a tangent line that has i)a zero slope ii) a minimum slope. Use diagrams to explain answer

Answer 1

The derivative of cos x is the negative of sin x. So cos x has zero slope whn sin x = 0 and cos x has a minium slope when sin x = +1. The interval 0 < x < 2 pi includes the end points. So there are three points at zero slope (0, pi, and 2 pi) and one minimum slope at 3 pi / 2.

Answer 2

y'=-sinx=0

y=cosx

x = sin^-1(0) = 0, pi, 2pi

three points on y=cosx, between x=0 and x=2pi,

have tangent lines with slopes = 0

(0,1), (pi,-1), (2pi,1)

minimum slope is when slope = -sinx =-1, sinx=1, x=sin^-1(1) = pi/2

the point is (pi/2, 0)

one "diagram" is the graph of cosx, use a graphing calculator

zero slope is when cosx is at a maximum or minimum

minimum slope is when the cosx curve is going down, the inflection point where it switches from concave down to concave up