Let `f(x)=3x^2-7x+c ,` where `c` is a variable coefficient and `x >7/6` . Then the value of `[c]` such that `f(x)` touches `f^(-1)(x)` is (where [.] represents greatest integer function)_________

Explanation

To find the value of \( c \) such that the function \( f(x) = 3x^2 - 7x + c \) touches its inverse \( f^{-1}(x) \) for \( x > \frac{7}{6} \), we follow these steps:

### Step 1: Understand the condition for tangency
For the function \( f(x) \) to touch its inverse \( f^{-1}(x) \), they must intersect at a point where the tangent to the curve is also equal to the line \( y = x \). This means we need to find the points where \( f(x) = x \).

### Step 2: Set up the equation
We set the function equal to \( x \):
\[ f(x) = x \]
This gives us:
\[ 3x^2 - 7x + c = x \]
Rearranging this, we obtain:
\[ 3x^2 - 8x + c = 0 \]

### Step 3: Condition for equal roots
For the quadratic equation \( 3x^2 - 8x + c = 0 \) to have equal roots, the discriminant must be zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
In our case:
- \( a = 3 \)
- \( b = -8 \)
- \( c = c \)

Thus, the discriminant becomes:
\[ D = (-8)^2 - 4 \cdot 3 \cdot c = 64 - 12c \]
Setting the discriminant equal to zero for equal roots:
\[ 64 - 12c = 0 \]

### Step 4: Solve for \( c \)
Now we solve for \( c \):
\[ 12c = 64 \]
\[ c = \frac{64}{12} = \frac{16}{3} \]

### Step 5: Find the greatest integer function value
Now, we need to find the greatest integer function value of \( c \):
\[ [c] = \left[\frac{16}{3}\right] \]
Calculating \( \frac{16}{3} \):
\[ \frac{16}{3} \approx 5.33 \]
Thus, the greatest integer less than or equal to \( \frac{16}{3} \) is:
\[ [c] = 5 \]

### Final Answer
The value of \([c]\) such that \( f(x) \) touches \( f^{-1}(x) \) is:
\[ \boxed{5} \]