If l,g are the least and greatest values of `9 cos 2theta - 24 cos theta - 20` than ( l ,g ) is equal to

Options

A. `( - 35,35)`
B. `( - 35,-13)`
C. `(-37,13)`
D. `( - 37,-35)`

Answer

To find the least (l) and greatest (g) values of the expression \(9 \cos 2\theta - 24 \cos \theta - 20\), we can follow these steps:

### Step 1: Rewrite the expression using the double angle identity
We know that \(\cos 2\theta = 2 \cos^2 \theta - 1\). Thus, we can rewrite the expression:

\[ 9 \cos 2\theta - 24 \cos \theta - 20 = 9(2 \cos^2 \theta - 1) - 24 \cos \theta - 20 \]

### Step 2: Simplify the expression
Now, we simplify the expression:

\[ = 18 \cos^2 \theta - 9 - 24 \cos \theta - 20 \]
\[ = 18 \cos^2 \theta - 24 \cos \theta - 29 \]

### Step 3: Let \(x = \cos \theta\)
Let \(x = \cos \theta\). The expression now becomes:

\[ f(x) = 18x^2 - 24x - 29 \]

### Step 4: Find the vertex of the quadratic
The vertex of a quadratic \(ax^2 + bx + c\) can be found using the formula \(x = -\frac{b}{2a}\). Here, \(a = 18\) and \(b = -24\):

\[ x = -\frac{-24}{2 \cdot 18} = \frac{24}{36} = \frac{2}{3} \]

### Step 5: Calculate the value of the function at the vertex
Now, we substitute \(x = \frac{2}{3}\) back into the expression to find the maximum or minimum value:

\[ f\left(\frac{2}{3}\right) = 18\left(\frac{2}{3}\right)^2 - 24\left(\frac{2}{3}\right) - 29 \]
\[ = 18 \cdot \frac{4}{9} - 24 \cdot \frac{2}{3} - 29 \]
\[ = 8 - 16 - 29 = -37 \]

### Step 6: Determine the range of \(x\)
Since \(x = \cos \theta\), the range of \(x\) is \([-1, 1]\).

### Step 7: Evaluate the function at the endpoints
Now, we evaluate \(f(x)\) at the endpoints \(x = -1\) and \(x = 1\):

1. For \(x = -1\):
\[ f(-1) = 18(-1)^2 - 24(-1) - 29 = 18 + 24 - 29 = 13 \]

2. For \(x = 1\):
\[ f(1) = 18(1)^2 - 24(1) - 29 = 18 - 24 - 29 = -35 \]

### Step 8: Identify the least and greatest values
From the calculations:
- The least value \(l = -37\) (at the vertex).
- The greatest value \(g = 13\) (at \(x = -1\)).

### Final Result
Thus, the least and greatest values are:

\[ (l, g) = (-37, 13) \]