Question #17673

Find the absolute maximum and minimum values of the following function over the given interval. If there are multiple points in a single category list the points in increasing order in x value and enter N in any blank that you don't need to use.
f(x)=(9cosx)/(14+7sinx), 0≤x≤2π
Absolute maxima
x = y =
x = y =
x = y =
Absolute minima
x = y =
x = y =
x = y =

Expert's answer

function is continuous so we can check only points x=0,x=2pi and x where f'=0 when x=0 and x=2pi we get the same value f(x)=9/14

f'(x)=-(9 (1+2 sin(x)))/(7 (2+sin(x))^2)

f'(x)=0 so 1+2sin(x)=0 so x=7pi/6, x=11pi/6

x=7pi/6, f(x)=-3sqrt(3)/7

x=11pi/6 , f(x)=-3sqrt(3)/7

so absolute maxima

x=0, y=9/14

x=2pi, y=9/14

absolute minima

x=7pi/6 y=-3sqrt(3)/7

x=11pi/6 y=-3sqrt(3)/7

f'(x)=-(9 (1+2 sin(x)))/(7 (2+sin(x))^2)

f'(x)=0 so 1+2sin(x)=0 so x=7pi/6, x=11pi/6

x=7pi/6, f(x)=-3sqrt(3)/7

x=11pi/6 , f(x)=-3sqrt(3)/7

so absolute maxima

x=0, y=9/14

x=2pi, y=9/14

absolute minima

x=7pi/6 y=-3sqrt(3)/7

x=11pi/6 y=-3sqrt(3)/7

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