Let `f(x) = cos^(2) x + cos x +3` then greatest value of f(x) + least value of f(x) is equal to

Answer

To solve the problem, we need to find the greatest and least values of the function \( f(x) = \cos^2 x + \cos x + 3 \) and then sum these values.

### Step-by-Step Solution:

1. **Define the Function**:
\[ f(x) = \cos^2 x + \cos x + 3 \]

2. **Differentiate the Function**:
To find the critical points, we first differentiate \( f(x) \) with respect to \( x \):
\[ f'(x) = \frac{d}{dx}(\cos^2 x) + \frac{d}{dx}(\cos x) + \frac{d}{dx}(3) \]
Using the chain rule:
\[ f'(x) = 2\cos x(-\sin x) + (-\sin x) + 0 = -2\cos x \sin x - \sin x \]
Factoring out \( -\sin x \):
\[ f'(x) = -\sin x(2\cos x + 1) \]

3. **Set the Derivative to Zero**:
To find the critical points, we set the derivative equal to zero:
\[ -\sin x(2\cos x + 1) = 0 \]
This gives us two cases:
- \( \sin x = 0 \)
- \( 2\cos x + 1 = 0 \)

4. **Solve for Critical Points**:
- From \( \sin x = 0 \):
\[ x = n\pi \quad (n \in \mathbb{Z}) \]
- From \( 2\cos x + 1 = 0 \):
\[ \cos x = -\frac{1}{2} \quad \Rightarrow \quad x = \frac{2\pi}{3} + 2n\pi \text{ or } x = \frac{4\pi}{3} + 2n\pi \]

5. **Evaluate \( f(x) \) at Critical Points**:
- For \( x = 0 \):
\[ f(0) = \cos^2(0) + \cos(0) + 3 = 1 + 1 + 3 = 5 \]
- For \( x = \frac{2\pi}{3} \):
\[ f\left(\frac{2\pi}{3}\right) = \cos^2\left(\frac{2\pi}{3}\right) + \cos\left(\frac{2\pi}{3}\right) + 3 \]
\[ = \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right) + 3 = \frac{1}{4} - \frac{1}{2} + 3 = \frac{1}{4} - \frac{2}{4} + \frac{12}{4} = \frac{11}{4} = 2.75 \]
- For \( x = \frac{4\pi}{3} \):
\[ f\left(\frac{4\pi}{3}\right) = \cos^2\left(\frac{4\pi}{3}\right) + \cos\left(\frac{4\pi}{3}\right) + 3 \]
\[ = \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right) + 3 = \frac{1}{4} - \frac{1}{2} + 3 = \frac{1}{4} - \frac{2}{4} + \frac{12}{4} = \frac{11}{4} = 2.75 \]

6. **Determine Maximum and Minimum Values**:
- The maximum value of \( f(x) \) is \( 5 \) (at \( x = 0 \)).
- The minimum value of \( f(x) \) is \( 2.75 \) (at \( x = \frac{2\pi}{3} \) and \( x = \frac{4\pi}{3} \)).

7. **Calculate the Sum of Maximum and Minimum Values**:
\[ \text{Greatest value} + \text{Least value} = 5 + 2.75 = 7.75 \]

### Final Answer:
The greatest value of \( f(x) \) plus the least value of \( f(x) \) is \( 7.75 \).

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