Verified answer

f(x) = Asin(Bx + C) + D

a)

Amplitude

IAI = I 3 I = 3

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b)

Period = 2π / B

Period = 2π / (1/2)

Period = 4π

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PHASE SHIFT:

C = - π/6

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x intercepts, means when y = 0

0 = 3 * sin( (1/2)x + π/6)

0 = sin( (1/2)x + π/6)

0 , π , 2π = (1/2)x + π/6

0 - π/6 , π - π/6 , 2π - π/6 = (1/2) x

- π/6 , 5π/6 , 11π/6 = (1/2) x

- 2π/6 , 10π/6 , 22π/6 = x

- π/3 , 5π/3 , 11π/3 = x

(- π/3 , 0) , (5π/3 , 0) , (11π/3 , 0)

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to find the max or min, we are going to find the first derivative and equal it to zero as the following:

f '(x) = 3 * cos( (1/2)x + π/6) * (1/2)

0 = cos((1/2)x + π/6)

π/2 , 3π/2 = (1/2)x + π/6

π/2 - π/6 , 3π/2 - π/6 = (1/2) x

2π/6 , 8π/6 = (1/2) x

π/3 , 4π/3 = x <---- critical points

Now, we need to plug these critical points into the original equation as the following:

f (π/3) = 3 * sin( π/3 + π/6) = 3 <---- maximum

f (4π/3) = 3 * sin( 4π/3 + π/6) = -3 <---- minimum

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Maximize[y == 3 Sin[1/(2 x) + Pi/6], x]

Minimize[y == 3 Sin[1/(2 x) + Pi/6], x]

lim_(x->±infinity) 3 sin(pi/6+1/(2 x)) = 3/2~~1.5