F.5 A.maths-Differentiation

Consider the curve y=f(x)=3^(1/2)sin2x+cos2x, where -90=< x =< 90.
(a) The curve y= f(x) intersects the coordinate axes. Find the coordinates of the points of intersection.
(b)Find the turning point(s) of the curve y=f(x) . For each point , test whether it is a maximum or a minimum point.
(c) Sketch the curve y=f(x).

(a)
let √3sin2x+cos2x=Rcos(2x-y)=Rcos2xcosy+Rsin2xsiny
Rcosy=1
Rsiny=√3
R=2, y=60
√3sin2x+cos2x=2cos(2x-60)
let √3sin2x+cos2x=0
2cos(2x-60)=0
2x-60=90 or 2x-60=-90
x=75 or -15
the coordinates of the points of intersection is (75,0) (-15,0)
(b)
dy/dx=d[2cos(2x-60)]/dx=-4sin(2x-60)
d^2y/dx^2=-8cos(2x-60)
let dy/dx=0
-4sin(2x-60)=0
2x-60=0 or 2x-60=-180
x=30 or -60
when x=30 d^2y/dx^2<0
when x=-60 d^2y/dx^2>0
so (30,2) is a maximum point while (-60,-2) is a minimum point
(c)
see http://hk.geocities.com/myisland8132/doc3.pub