Let `f : R to R` be a function defined as `f(x) = {((b + 2)x-11c, x < -1),(x^2 + bx + c, -1lexle1),(2bx - 7x, 1 < x le2),(9/4, x>2):}` then f(x)is:

Options

A. Continuous if `b = -1 and c = -1/4`
B. Continuous if `b = 1/8 and c = 1/4`
C. Continuous if `b = 11/25 and c = 7/100`
D. Not continuous for any values of b and c

Explanation

To determine the function \( f(x) \) given in the piecewise form and check for continuity, we will analyze the function at the points where the pieces meet, specifically at \( x = -1 \), \( x = 1 \), and \( x = 2 \).

### Step 1: Analyze continuity at \( x = -1 \)

The function is defined as:
- For \( x < -1 \): \( f(x) = (b + 2)x - 11c \)
- For \( -1 \leq x \leq 1 \): \( f(x) = x^2 + bx + c \)

To ensure continuity at \( x = -1 \), we need:
\[ \lim_{x \to -1^-} f(x) = \lim_{x \to -1^+} f(x) \]

Calculating the left-hand limit:
\[ \lim_{x \to -1^-} f(x) = (b + 2)(-1) - 11c = -b - 2 - 11c \]

Calculating the right-hand limit:
\[ \lim_{x \to -1^+} f(x) = (-1)^2 + b(-1) + c = 1 - b + c \]

Setting the two limits equal for continuity:
\[ -b - 2 - 11c = 1 - b + c \]

### Step 2: Simplify the equation

Rearranging gives:
\[ -b - 2 - 11c = 1 - b + c \implies -2 - 11c = 1 + c \implies -12c = 3 \implies c = -\frac{1}{4} \]

### Step 3: Analyze continuity at \( x = 1 \)

Next, we check continuity at \( x = 1 \):
- For \( -1 \leq x \leq 1 \): \( f(x) = x^2 + bx + c \)
- For \( 1 < x \leq 2 \): \( f(x) = 2bx - 7x \)

Setting the limits equal:
\[ \lim_{x \to 1^-} f(x) = 1 + b - \frac{1}{4} = 1 + b - 0.25 = b + 0.75 \]
\[ \lim_{x \to 1^+} f(x) = 2b(1) - 7(1) = 2b - 7 \]

Setting these equal for continuity:
\[ b + 0.75 = 2b - 7 \]

### Step 4: Solve for \( b \)

Rearranging gives:
\[ 0.75 + 7 = 2b - b \implies 7.75 = b \]

### Step 5: Analyze continuity at \( x = 2 \)

Finally, we check continuity at \( x = 2 \):
- For \( 1 < x \leq 2 \): \( f(x) = 2bx - 7x \)
- For \( x > 2 \): \( f(x) = \frac{9}{4} \)

Setting the limits equal:
\[ \lim_{x \to 2^-} f(x) = 2b(2) - 7(2) = 4b - 14 \]
\[ \lim_{x \to 2^+} f(x) = \frac{9}{4} \]

Setting these equal for continuity:
\[ 4b - 14 = \frac{9}{4} \]

### Step 6: Solve for \( b \)

Multiplying through by 4 to eliminate the fraction:
\[ 16b - 56 = 9 \implies 16b = 65 \implies b = \frac{65}{16} \]

### Conclusion

We have found:
- \( b = \frac{65}{16} \)
- \( c = -\frac{1}{4} \)

Thus, the function \( f(x) \) is continuous for the values of \( b \) and \( c \) found.