Find the critical numbers of the function on the interval 0 ≤ θ &lt; 2 π . f ( θ ) = 2 cos ⁡ ( θ ) + ( sin ⁡ ( θ ) ) 2 θ = ? (smallervalue) θ = ? (larger value)

Question

Find the critical numbers of the function on the interval   0  ≤  θ  &amp;lt;  2  π  .  f  (  θ  )  =  2  cos  ⁡  (  θ  )  +  (  sin  ⁡  (  θ  )      )    2    θ  =  ? (smallervalue)  θ  =  ? (larger value)

Smeawnswelapno · Accepted Answer

f  (  θ  )  =  2  cos  ⁡  (  θ  )  +  (  sin  ⁡  (  θ  )      )    2        f    ′    (  θ  )  =  2  −  sin  ⁡  (  θ  )  +  2  sin  ⁡  (  θ  )  cos  ⁡  (  θ  )      f    ′    (  θ  )  =  2  −  sin  ⁡  (  θ  )  [  1  −  cos  ⁡  θ  ]Set       f    ′    (  θ  )  =  0 and solve for   θ to find critical numbers  0  =  −  2  sin  ⁡  (  θ  )  [  1  −  cos  ⁡  (  θ  )  ]    −  2  sin  ⁡  θ  =  0  ;     1  −  cos  ⁡  θ  =  0    sin  ⁡  θ  =  0  ;     cos  ⁡  θ  =  1    θ  =  0  ;     θ  =  π  ;     θ  =  2  π==&amp;gt; on the interval   0  ≤  θ  &amp;lt;  2  π we have  θ  =  0  ;  θ  =  πTherefore  θ  =  0 Small value  θ  =  π Large value

Find the critical numbers of the function on the interval 0<= theta < 2pi. f(theta) = 2cos(theta) +(sin(theta))^2 theta =? (smallervalue) theta =? (larger value)

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