Find stationary points in interval 0 <= x < 2π and find out whether they are maxima, minima or neither.
(a) y= sin2 x + 2cos x
(b) y= cos2x + x
(c) y = sin x + cos x
please show your working with the final answer---So i know the method
... thanks
to find whether maxima, minima or inflexion u can use d2y/dx2
(a) dy/dx = 2 sin x cos x - 2 sin x = 2 sin x ( cos x - 1)
when dy/dx = 0,
sin x = 0, or cos x = 1
x = 0, π, 2π
when x = 0, y = 2
when x = π, y = -2
when x = 2π, y = 2
dy/dx = sin 2x - 2 sin x
d²y/dx² = 2 sin 2x cos 2x - 2 cos x
x = 0, d²y/dx² = -2 < 0 ==> maxima
x = π, d²y/dx² = 2 > 0 ==> minima
x = π, d²y/dx² = -2 < 0 ==> maxima
Alternatively, you could use your graphic calculator to solve for everything
Try yourself for (b) and (c) using the same methods
dy/dx = cos x - sin x (when dy/dx=0)
cos x - sin x = 0
I just need to know: how to solve for x??
Hi,
When dy/dx = 0, we have 2 (sin x) (cos x - 1) = 0.
So sin x = 0 or cos x - 1 = 0, since we need at least a single zero to make the equation valid.
Your method is alright, although it is good to know the special angles for x that would give sin x = 0 and cos x = 1.
Since cos x - sin x = 0, sin x = cos x. This implies that tan x = 1, provided that cos x is never zero.
Thanks!
Cheers,
Wen Shih