The number of solutions of `12 cos^(3) x-7 cos^(2) x+4 cos x=9` is

Options

A. 0
B. 2
C. infinite
D. none of these

Explanation

To solve the equation \(12 \cos^3 x - 7 \cos^2 x + 4 \cos x = 9\), we will follow these steps:

### Step 1: Rearranging the Equation
First, we will move all terms to one side of the equation:
\[ 12 \cos^3 x - 7 \cos^2 x + 4 \cos x - 9 = 0 \]

### Step 2: Substituting \(y = \cos x\)
Let \(y = \cos x\). Then, we can rewrite the equation as:
\[ 12y^3 - 7y^2 + 4y - 9 = 0 \]

### Step 3: Analyzing the Cubic Equation
Now we have a cubic equation in terms of \(y\):
\[ 12y^3 - 7y^2 + 4y - 9 = 0 \]
We need to find the number of real solutions for this cubic equation.

### Step 4: Finding the Derivative
To analyze the number of solutions, we can find the derivative of the cubic function:
\[ f(y) = 12y^3 - 7y^2 + 4y - 9 \]
The derivative is:
\[ f'(y) = 36y^2 - 14y + 4 \]

### Step 5: Finding Critical Points
Set the derivative equal to zero to find critical points:
\[ 36y^2 - 14y + 4 = 0 \]
Using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ y = \frac{14 \pm \sqrt{(-14)^2 - 4 \cdot 36 \cdot 4}}{2 \cdot 36} \]
\[ y = \frac{14 \pm \sqrt{196 - 576}}{72} \]
\[ y = \frac{14 \pm \sqrt{-380}}{72} \]
Since the discriminant is negative, there are no real roots for the derivative, indicating that the cubic function is either strictly increasing or strictly decreasing.

### Step 6: Analyzing the Behavior of the Cubic Function
Since \(f'(y)\) has no real roots, \(f(y)\) is strictly increasing. Therefore, it can cross the x-axis at most once.

### Step 7: Evaluating the Limits
Next, we evaluate the limits of \(f(y)\) as \(y\) approaches the boundaries of the cosine function:
- As \(y \to -1\):
\[ f(-1) = 12(-1)^3 - 7(-1)^2 + 4(-1) - 9 = -12 - 7 - 4 - 9 = -32 \]
- As \(y \to 1\):
\[ f(1) = 12(1)^3 - 7(1)^2 + 4(1) - 9 = 12 - 7 + 4 - 9 = 0 \]

### Step 8: Conclusion on the Number of Solutions
Since \(f(-1) < 0\) and \(f(1) = 0\), and \(f(y)\) is strictly increasing, there is exactly one solution for \(y\) in the interval \([-1, 1]\).

### Step 9: Finding Solutions for \(x\)
Since \(y = \cos x\) has one solution, we can find the corresponding values of \(x\):
\[ \cos x = y \implies x = \cos^{-1}(y) + 2n\pi \text{ or } x = -\cos^{-1}(y) + 2n\pi, \quad n \in \mathbb{Z} \]
This indicates that there are infinitely many solutions for \(x\).

### Final Answer
Thus, the number of solutions for the original equation is **infinite**.

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