Consider the equation below. f(x) = 7 cos 2 x 14 sin x, 0 x 2 (a) Find the inter

Question

Consider the equation below.
f(x) = 7 cos2x 14 sin x, 0 x 2
(a) Find the interval on which f is increasing. (Enter your answer in interval notation.)

Find the interval on which f is decreasing. (Enter your answer in interval notation.)

(b) Find the local minimum and maximum values of f.
local minimum    
local maximum    

(c) Find the inflection points.
(x, y) =

(smaller x-value)
(x, y) =

(larger x-value)

Find the interval on which f is concave up. (Enter your answer in interval notation.)

Find the interval on which f is concave down. (Enter your answer in interval notation.)

Explanation / Answer

f(x) = 7 cos2x - 14 sin x

f'(x) = 7*2 * (-sin 2x) - 14*cos x
      = -14*sin 2x - 14 * cos x
      = -14 (sin 2x + cos x)
  
f''(x) = -14*(2*cos 2x - sin x)
       = -28*cos 2x + 14*sin x
  
put f'(x)= 0
-14 (sin 2x + cos x) = 0
sin 2x = - cos x
2*sin x* cos x = - cos x
sin x = -1/2
x = 2*pi - pi/6 = 11*pi/6
x = pi + pi/6 = 7*pi/6

at x = 11*pi/6
f''(x) = -28*cos 2x + 14*sin x
       = -28* cos (11*pi/6) + 14*sin (11*pi/6)
       = -31.2
       = negative
     
so, x = 11*pi/6 is maxima

at x = 7*pi/6
f''(x) = -28*cos 2x + 14*sin x
       = -28* cos (7*pi/6) + 14*sin (7*pi/6)
       = 17.25
       = positive
     
so, x = 7*pi/6 is minima

a)
f(x) is increasing for (7*pi/6,11*pi/6)        <----Answer 1
f(x) is decreasing for [0,7*pi/6)U(11*pi/6,2*pi]    <----Answer 2

b)
minima at x = 7*pi/6    <----Answer 3
maxima at x = 11*pi/6    <----Answer 4

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