Question

Find the critical numbers of the function.

33.\(f(\theta ) = 2\cos \theta + {\sin ^2}\theta \).

Short answer

The critical numbers of the function \(f(\theta ) = 2\cos \theta + {\sin ^2}\theta \) are in the form of\(\theta = n\pi ,\forall n = 0,1,2,3 \ldots ..\).

Step-by-step solution

Given data

The function is \(f(\theta ) = 2\cos \theta + {\sin ^2}\theta \).

Concept of critical number

A critical number of a function\(f\)is a number\(c\), if it satisfies either of the below conditions:

(1)\({f^\prime }(c) = 0\)

(2)\({f^\prime }(c)\), Does not exist.

Obtain the first derivative of the given function

Obtain the first derivative of the given function.

\(\begin{aligned}{c}{f^\prime }(\theta ) &= \frac{d}{{d\theta }}\left( {2\cos \theta + {{\sin }^2}\theta } \right)\\ &= \frac{d}{{d\theta }}(2\cos \theta ) + \frac{d}{{d\theta }}\left( {{{\sin }^2}\theta } \right)\\ &= 2( - \sin \theta ) + (2\sin \theta \cos \theta )\\ &= 2\sin \theta (\cos \theta - 1)\end{aligned}\)

Take \({f^\prime }(\theta ) = 0\)and obtain the critical numbers.

\(\begin{aligned}{c}2\sin \theta (\cos \theta - 1) &= 0\\\sin \theta &= 0\\\cos \theta &= 1\end{aligned}\)

Therefore, \(\sin \theta = 0\) and \(\cos \theta = 1\).

Compute the value of \(\theta \)

Compute the value of \(\theta \) for the two possible cases.

Recall the fact that the general solution of the equation \(\sin \theta = \sin \beta \):

\(\theta = n\pi + {( - 1)^n}\beta \) ……. (1)

And the general solution of the equation \(\cos \theta = \cos \beta \):

\(\theta = 2n\pi \pm \beta \) ……. (2)

Find the critical number of the function \(f(\theta ) = 2\cos \theta  + {\sin ^2}\theta \)

\(Case(1):\)

Consider, \(\sin \theta = 0\). ……. (3)

Express the equation (3).

\(\sin \theta = \sin (0)\) ……. (4)

Apply the general solution as shown in equation (1) and obtain the solution for equation (4).

\(\begin{aligned}{c}\theta &= n\pi + {( - 1)^n}(0)\quad \\({\rm{ }}\beta &= 0)\\\theta &= n\pi \end{aligned}\)

Therefore, the general solution of equation (3) is \(\theta = n\pi \).

Case (2):

Consider, \(\cos \theta = 1\). ……. (5)

Express the equation (5).

\(\cos \theta = \cos (0)\). ……. (6)

Apply the general solution as shown in equation (2) and obtain the solution for equation (6).

\(\begin{aligned}{c}\theta &= 2n\pi \pm 0\\\beta &= 0\\\theta &= 2n\pi \end{aligned}\)

Hence, the critical numbers are \(n\pi \)and \(2n\pi \).

Since, the \(2n\pi \) is the subset of \(n\pi \), take the critical number as \(n\pi \).

Thus, the critical numbers of the function \(f(\theta ) = 2\cos \theta + {\sin ^2}\theta \) are in the form of \(\theta = n\pi ,\forall n = 0,1,2,3 \ldots ..\)