Question

Find the critical numbers of the function.

45. \(f\left( \theta \right) = {\bf{2cos}}\theta + {\bf{si}}{{\bf{n}}^2}\theta \)

Short answer

The critical numbers obtained are \(\theta = n\pi ,\,\,\,\,\,\,\forall n \in {\rm I}\).

Step-by-step solution

Critical Numbers of a function

TheCritical numbers for any function \(f\left( \theta \right)\) are obtained by putting \(f'\left( \theta \right) = 0\).

Differentiating the function for critical numbers:

The given function is:

\(f\left( \theta \right) = 2\cos \theta + {\sin ^2}\theta \)

On differentiating with respect to\(\theta \)as:

\(\begin{aligned}{c}f'\left( \theta \right) &= \frac{d}{{d\theta }}\left( {2\cos \theta + {{\sin }^2}\theta } \right)\\ &= 2\frac{d}{{d\theta }}\left( {\cos \theta } \right) + \frac{d}{{d\theta }}\left( {{{\sin }^2}\theta } \right)\\ &= - 2\sin \theta + \sin 2\theta \end{aligned}\)

Solving for critical numbers:

\(\begin{aligned}{c}f'\left( \theta \right) &= 0\\ - 2\sin \theta + \sin 2\theta &= 0\\2\sin \theta \left( {\cos \theta - 1} \right) &= 0\\\sin \theta &= 0,\,\,\,\,\cos \theta = 1\\\theta &= n\pi ,\,\,\,\,\theta = 2n\pi ,\,\,\,\,\,\,\forall n \in {\rm I}\\\theta &= n\pi ,\,\,\,\,\,\,\forall n \in {\rm I}\end{aligned}\)

Hence, the critical numbers are \(\theta = n\pi ,\,\,\,\,\,\,\forall n \in {\rm I}\).