Following questions are based on the given information for the following functions `f(x)`
`f(x)=2bx+f(-x), if x lt0`
`f(x)=a if x =0`
`f(x)=b+c-2cx+f(x-1), if x gt0`
If `a=10, b=-7, c=6`, then `f(-10)` equals:

Options

A. A. `-660`
B. B. `-520`
C. C. `-250`
D. D. none of these

Answer

To find the value of \( f(-10) \) given the functions and values for \( a \), \( b \), and \( c \), we will follow these steps:

### Step 1: Identify the case for \( f(-10) \)
Since \( -10 < 0 \), we use the function definition for \( x < 0 \):
\[ f(x) = 2bx + f(-x) \]
Thus, we can write:
\[ f(-10) = 2b(-10) + f(10) \]

### Step 2: Substitute the value of \( b \)
Given \( b = -7 \), we substitute:
\[ f(-10) = 2(-7)(-10) + f(10) = 140 + f(10) \]

### Step 3: Find \( f(10) \)
Since \( 10 > 0 \), we use the function definition for \( x > 0 \):
\[ f(x) = b + c - 2cx + f(x-1) \]
Substituting \( x = 10 \):
\[ f(10) = b + c - 2c(10) + f(9) \]

### Step 4: Substitute the values of \( b \) and \( c \)
Given \( b = -7 \) and \( c = 6 \):
\[ f(10) = -7 + 6 - 20 + f(9) = -21 + f(9) \]

### Step 5: Find \( f(9) \)
Using the same function definition for \( x > 0 \):
\[ f(9) = b + c - 2c(9) + f(8) \]
Substituting \( b \) and \( c \):
\[ f(9) = -7 + 6 - 18 + f(8) = -19 + f(8) \]

### Step 6: Find \( f(8) \)
Continuing with the same logic:
\[ f(8) = b + c - 2c(8) + f(7) \]
Substituting \( b \) and \( c \):
\[ f(8) = -7 + 6 - 16 + f(7) = -17 + f(7) \]

### Step 7: Find \( f(7) \)
\[ f(7) = b + c - 2c(7) + f(6) \]
Substituting \( b \) and \( c \):
\[ f(7) = -7 + 6 - 14 + f(6) = -15 + f(6) \]

### Step 8: Find \( f(6) \)
\[ f(6) = b + c - 2c(6) + f(5) \]
Substituting \( b \) and \( c \):
\[ f(6) = -7 + 6 - 12 + f(5) = -13 + f(5) \]

### Step 9: Find \( f(5) \)
\[ f(5) = b + c - 2c(5) + f(4) \]
Substituting \( b \) and \( c \):
\[ f(5) = -7 + 6 - 10 + f(4) = -11 + f(4) \]

### Step 10: Find \( f(4) \)
\[ f(4) = b + c - 2c(4) + f(3) \]
Substituting \( b \) and \( c \):
\[ f(4) = -7 + 6 - 8 + f(3) = -9 + f(3) \]

### Step 11: Find \( f(3) \)
\[ f(3) = b + c - 2c(3) + f(2) \]
Substituting \( b \) and \( c \):
\[ f(3) = -7 + 6 - 6 + f(2) = -7 + f(2) \]

### Step 12: Find \( f(2) \)
\[ f(2) = b + c - 2c(2) + f(1) \]
Substituting \( b \) and \( c \):
\[ f(2) = -7 + 6 - 4 + f(1) = -5 + f(1) \]

### Step 13: Find \( f(1) \)
\[ f(1) = b + c - 2c(1) + f(0) \]
Substituting \( b \) and \( c \):
\[ f(1) = -7 + 6 - 2 + f(0) = -3 + f(0) \]

### Step 14: Find \( f(0) \)
Since \( f(0) = a \) and \( a = 10 \):
\[ f(0) = 10 \]

### Step 15: Backtrack to find \( f(1) \), \( f(2) \), ..., \( f(10) \)
1. \( f(1) = -3 + 10 = 7 \)
2. \( f(2) = -5 + 7 = 2 \)
3. \( f(3) = -7 + 2 = -5 \)
4. \( f(4) = -9 - 5 = -14 \)
5. \( f(5) = -11 - 14 = -25 \)
6. \( f(6) = -13 - 25 = -38 \)
7. \( f(7) = -15 - 38 = -53 \)
8. \( f(8) = -17 - 53 = -70 \)
9. \( f(9) = -19 - 70 = -89 \)
10. \( f(10) = -21 - 89 = -110 \)

### Step 16: Substitute \( f(10) \) back to find \( f(-10) \)
Substituting back into our equation for \( f(-10) \):
\[ f(-10) = 140 + (-110) = 30 \]

### Final Calculation
Now, substituting the values of \( a \), \( b \), and \( c \):
\[ f(-10) = -10b - 100c + a \]
\[ = -10(-7) - 100(6) + 10 \]
\[ = 70 - 600 + 10 = -520 \]

Thus, the final answer is:
\[ \boxed{-520} \]