consider the function on the interval (0,2pi)

Question

Consider the function on the interval (0, 2π).

f(x) = sin(x) cos(x) + 9

(a) Find the open interval(s) on which the function is increasing or decreasing. (Enter your answers using interval notation.)

increasing ______

decreasing ______

(b) Apply the First Derivative Test to identify all relative extrema.

relative maxima (x, y) = (____) (smaller x-value)

(x, y) = (____) (larger x-value)

relative minima (x, y) = (____) (smaller x-value)

(x, y) = (____) (larger x-value)

Answer 1

Per product rule the derivative is sin * (-sin) + cos * cos = cos^2 - sin^2 = cos(2x)

cosine is zero at 90 and 270 degrees......

2x = pi/2 or 2x = 3*pi/2

x = pi/4=45 or x = 3*pi/4=135

min max radians cos 2x

0 45 0 0.785398163 1.0000000 0.0000000

45 90 0.785398163 1.570796327 0.0000000 -1.0000000

90 135 1.570796327 2.35619449 -1.0000000 0.0000000

135 180 2.35619449 3.141592654 0.0000000 1.0000000

180 225 3.141592654 3.926990817 1.0000000 0.0000000

225 270 3.926990817 4.71238898 0.0000000 -1.0000000

270 315 4.71238898 5.497787144 -1.0000000 0.0000000

315 360 5.497787144 6.283185307 0.0000000 1.0000000