Answer:
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
