If `f(x)=[x]^({x}) +{x}^([x]) + sin (pix)` , Where [.] and {.} represent the greatest integer function and the fractional part function respectively, then `f'(7/2)` Is equal to

Options

A. `sqrt3In3+pi/4`
B. `sqrt3In3+(3pi)/4`
C. `sqrt3In3+pi+3/4`
D. `sqrt3In3+3/4`

Answer

To find \( f'(7/2) \) for the function

\[ f(x) = [x]^{\{x\}} + \{x\}^{[x]} + \sin(\pi x) \]

where \([x]\) is the greatest integer function and \(\{x\}\) is the fractional part function, we will follow these steps:

### Step 1: Identify the values of \([x]\) and \(\{x\}\) at \(x = \frac{7}{2}\)

Since \( \frac{7}{2} = 3.5 \):
- The greatest integer function \([x]\) gives us \([3.5] = 3\).
- The fractional part function \(\{x\}\) gives us \(\{3.5\} = 3.5 - 3 = 0.5\).

### Step 2: Substitute these values into the function \(f(x)\)

Now substituting these values into the function:

\[ f\left(\frac{7}{2}\right) = [3.5]^{\{3.5\}} + \{3.5\}^{[3.5]} + \sin\left(\pi \cdot \frac{7}{2}\right) \]

This simplifies to:

\[ f\left(\frac{7}{2}\right) = 3^{0.5} + 0.5^{3} + \sin\left(\frac{7\pi}{2}\right) \]

### Step 3: Calculate each term

1. \(3^{0.5} = \sqrt{3}\)
2. \(0.5^{3} = \frac{1}{8}\)
3. \(\sin\left(\frac{7\pi}{2}\right)\):
- Since \(\frac{7\pi}{2} = 3.5\pi\), and \(3.5\pi\) is an odd multiple of \(\frac{\pi}{2}\), we have \(\sin(3.5\pi) = -1\).

Thus,

\[ f\left(\frac{7}{2}\right) = \sqrt{3} + \frac{1}{8} - 1 \]

### Step 4: Combine the terms

Combining these gives:

\[ f\left(\frac{7}{2}\right) = \sqrt{3} - \frac{7}{8} \]

### Step 5: Differentiate \(f(x)\)

Now we need to find \(f'(x)\):

\[ f'(x) = \frac{d}{dx}\left([x]^{\{x\}} + \{x\}^{[x]} + \sin(\pi x)\right) \]

Using the chain rule and product rule, we differentiate each term:

1. For \( [x]^{\{x\}} \):
- This is a constant when \(x\) is in the interval \([3, 4)\), so the derivative is \(0\).

2. For \(\{x\}^{[x]}\):
- This can be differentiated using the power rule:
\[ \frac{d}{dx} \left(\{x\}^{3}\right) = 3\{x\}^{2} \cdot \frac{d}{dx}(\{x\}) = 3\{x\}^{2} \cdot 1 = 3\{x\}^{2} \]
where \(\{x\} = x - 3\).

3. For \(\sin(\pi x)\):
- The derivative is \(\pi \cos(\pi x)\).

Thus, combining these gives:

\[ f'(x) = 0 + 3(x - 3)^{2} + \pi \cos(\pi x) \]

### Step 6: Evaluate \(f'(7/2)\)

Substituting \(x = \frac{7}{2}\):

\[ f'\left(\frac{7}{2}\right) = 3\left(\frac{7}{2} - 3\right)^{2} + \pi \cos\left(\frac{7\pi}{2}\right) \]

Calculating each term:

1. \(\frac{7}{2} - 3 = \frac{1}{2}\), so:
\[ 3\left(\frac{1}{2}\right)^{2} = 3 \cdot \frac{1}{4} = \frac{3}{4} \]

2. \(\cos\left(\frac{7\pi}{2}\right) = 0\) (since it's an odd multiple of \(\frac{\pi}{2}\)).

Thus,

\[ f'\left(\frac{7}{2}\right) = \frac{3}{4} + 0 = \frac{3}{4} \]

### Final Answer

The value of \(f'\left(\frac{7}{2}\right)\) is

\[ \frac{3}{4} \]