Let f be a function defined on `[-(pi)/(2), (pi)/(2)]` by f(x) = `3 cos^(4) x-6 cos^(3) x - 6 cos^(2) x-3`. Then the range of f(x) is

Options

A. `[-12, -3]`
B. `[-6, -3]`
C. `[-6, 3]`
D. `(-12, 3]`

Answer

To find the range of the function \( f(x) = 3 \cos^4 x - 6 \cos^3 x - 6 \cos^2 x - 3 \) defined on the interval \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), we will follow these steps:

### Step 1: Substitute \( \cos x \) with \( t \)
Let \( t = \cos x \). Since \( x \) is in the interval \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), \( t \) will vary from 0 to 1. Thus, we can rewrite the function as:
\[ f(t) = 3t^4 - 6t^3 - 6t^2 - 3 \]

### Step 2: Find the derivative of \( f(t) \)
To find the critical points, we need to differentiate \( f(t) \):
\[ f'(t) = 12t^3 - 18t^2 - 12t \]

### Step 3: Set the derivative to zero
Now, we set the derivative equal to zero to find the critical points:
\[ 12t^3 - 18t^2 - 12t = 0 \]
Factoring out \( 6t \):
\[ 6t(2t^2 - 3t - 2) = 0 \]
This gives us one solution:
\[ t = 0 \]
Now, we solve the quadratic equation \( 2t^2 - 3t - 2 = 0 \) using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-2)}}{2 \cdot 2} \]
\[ t = \frac{3 \pm \sqrt{9 + 16}}{4} = \frac{3 \pm 5}{4} \]
This gives us:
\[ t = 2 \quad \text{(not valid since } t \in [0, 1]\text{)} \quad \text{and} \quad t = -\frac{1}{2} \quad \text{(not valid since } t \in [0, 1]\text{)} \]

### Step 4: Evaluate the function at the endpoints and critical points
We will evaluate \( f(t) \) at the valid critical point \( t = 0 \) and at the endpoints \( t = 0 \) and \( t = 1 \):
1. \( f(0) = 3(0)^4 - 6(0)^3 - 6(0)^2 - 3 = -3 \)
2. \( f(1) = 3(1)^4 - 6(1)^3 - 6(1)^2 - 3 = 3 - 6 - 6 - 3 = -12 \)

### Step 5: Determine the range
From the evaluations:
- At \( t = 0 \), \( f(0) = -3 \)
- At \( t = 1 \), \( f(1) = -12 \)

Since \( f(t) \) is a continuous function on the interval \( [0, 1] \) and we have found the minimum and maximum values, the range of \( f(x) \) is:
\[ [-12, -3] \]

### Final Answer
The range of \( f(x) \) is \( [-12, -3] \).