Find two values of c in ( - π / 4, π / 4) such that f (c) is equal to the average value of f (x) = 2 cos (2x) on ( - π / 4, π / 4). Round your answers to three decimal places.

c₁ = 1/2 cos⁻¹ (2/π) = 0.44

c₂ = - 1/2 cos⁻¹ (2/π) = - 0.44

Step-by-step explanation:

the average value of f (x) = 2 cos (2x) on ( - π / 4, π / 4) is

av f (x) = ∫[2*cos (2x) ] dx / (∫dx) between limits of integration - π / 4 and π / 4

thus

av f (x) = ∫[cos (2x) ] dx / (∫dx) = [sin (2 * π / 4) - sin (2 * ( - π / 4) ] / [ π / 4 - (-π / 4) ]

= 2*sin (π/2) / (π/2) = 4/π

then the average value of f (x) is 4/π. Thus the values of c such that f (c) = av f (x) are

4/π = 2 cos (2c)

2/π = cos (2c)

c = 1/2 cos⁻¹ (2/π) = 0.44

c = 0.44

since the cosine function is symmetrical with respect to the y axis then also c = - 0.44 satisfy the equation

thus

c₁ = 1/2 cos⁻¹ (2/π) = 0.44

c₂ = - 1/2 cos⁻¹ (2/π) = - 0.44