The difference between the greatest and the least possible value of the expression `3-cosx +sin^2x` is

Options

A. `13/4`
B. `17/4`
C. `9/4`
D. `1/4`

Answer

To find the difference between the greatest and the least possible value of the expression \(3 - \cos x + \sin^2 x\), we can follow these steps:

### Step 1: Define the function
Let \( f(x) = 3 - \cos x + \sin^2 x \).

### Step 2: Differentiate the function
To find the critical points, we first differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}(3 - \cos x + \sin^2 x) = \sin x + 2\sin x \cos x \]
This simplifies to:
\[ f'(x) = \sin x (1 + 2\cos x) \]

### Step 3: Set the derivative to zero
To find the critical points, set \( f'(x) = 0 \):
\[ \sin x (1 + 2\cos x) = 0 \]
This gives us two cases:
1. \( \sin x = 0 \)
2. \( 1 + 2\cos x = 0 \)

### Step 4: Solve for \( x \)
For \( \sin x = 0 \):
- \( x = n\pi \) where \( n \) is an integer. The relevant values in the interval \( [0, 2\pi] \) are \( x = 0 \) and \( x = \pi \).

For \( 1 + 2\cos x = 0 \):
- \( \cos x = -\frac{1}{2} \)
- This occurs at \( x = \frac{2\pi}{3} \) and \( x = \frac{4\pi}{3} \).

### Step 5: Evaluate the function at critical points
Now we evaluate \( f(x) \) at the critical points \( x = 0, \pi, \frac{2\pi}{3} \):

1. For \( x = 0 \):
\[ f(0) = 3 - \cos(0) + \sin^2(0) = 3 - 1 + 0 = 2 \]

2. For \( x = \pi \):
\[ f(\pi) = 3 - \cos(\pi) + \sin^2(\pi) = 3 - (-1) + 0 = 4 \]

3. For \( x = \frac{2\pi}{3} \):
\[ f\left(\frac{2\pi}{3}\right) = 3 - \cos\left(\frac{2\pi}{3}\right) + \sin^2\left(\frac{2\pi}{3}\right) \]
Here, \( \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \) and \( \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} \):
\[ f\left(\frac{2\pi}{3}\right) = 3 + \frac{1}{2} + \left(\frac{\sqrt{3}}{2}\right)^2 = 3 + \frac{1}{2} + \frac{3}{4} = 3 + \frac{2}{4} + \frac{3}{4} = 3 + \frac{5}{4} = \frac{17}{4} \]

### Step 6: Identify the maximum and minimum values
From the evaluations:
- Minimum value: \( f(0) = 2 \)
- Maximum value: \( f\left(\frac{2\pi}{3}\right) = \frac{17}{4} \)

### Step 7: Calculate the difference
The difference between the greatest and least values is:
\[ \text{Difference} = \frac{17}{4} - 2 = \frac{17}{4} - \frac{8}{4} = \frac{9}{4} \]

### Final Answer
The difference between the greatest and the least possible value of the expression \(3 - \cos x + \sin^2 x\) is \(\frac{9}{4}\).

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