Math Snap

STEP 1

1. We are given the function $f(\theta) = 4 \cos(\theta) + 2 \sin^2(\theta)$.
2. We need to find the critical numbers of this function within the interval $-\pi \leq \theta \leq \pi$.
3. Critical numbers occur where the derivative is zero or undefined.

STEP 2

1. Find the derivative of $f(\theta)$.
2. Set the derivative equal to zero and solve for $\theta$.
3. Determine if the derivative is undefined for any $\theta$ in the given interval.
4. List the critical numbers in increasing order.

STEP 3

Find the derivative of $f(\theta)$.
$$f(\theta) = 4 \cos(\theta) + 2 \sin^2(\theta)$$ Use the chain rule and power rule to differentiate:
$$f'(\theta) = \frac{d}{d\theta}[4 \cos(\theta)] + \frac{d}{d\theta}[2 \sin^2(\theta)]$$ $$f'(\theta) = -4 \sin(\theta) + 4 \sin(\theta) \cos(\theta)$$

STEP 4

Set the derivative equal to zero:
$$-4 \sin(\theta) + 4 \sin(\theta) \cos(\theta) = 0$$ Factor out $4 \sin(\theta)$:
$$4 \sin(\theta)(-1 + \cos(\theta)) = 0$$ This gives us two equations to solve:
1. $\sin(\theta) = 0$
2. $-1 + \cos(\theta) = 0$

STEP 5

Solve $\sin(\theta) = 0$:
$$\theta = n\pi$$ Within the interval $-\pi \leq \theta \leq \pi$, the solutions are:
$$\theta = -\pi, 0, \pi$$

STEP 6

Solve $-1 + \cos(\theta) = 0$:
$$\cos(\theta) = 1$$ Within the interval $-\pi \leq \theta \leq \pi$, the solution is:
$$\theta = 0$$

STEP 7

Determine if the derivative is undefined for any $\theta$ in the given interval. The derivative is defined for all $\theta$ in the interval because it is composed of sine and cosine functions, which are defined everywhere.

List the critical numbers in increasing order:
$$\theta = -\pi, 0, \pi$$ The critical numbers of the function $f(\theta)$ are:
$$\boxed{-\pi, 0, \pi}$$