If the function f(x ) = `{{:(10 , x lt=3),(ax+b,3lt xlt 7),(18,x gt=7):}` is continuous function then the value of a + b is

Options

A. 8
B. 6
C. 4
D. 2

Answer

To solve the problem, we need to ensure that the function \( f(x) \) is continuous at the points where the definition of the function changes, specifically at \( x = 3 \) and \( x = 7 \).

The function is defined as follows:

\[ f(x) = \begin{cases} 10 & \text{if } x \leq 3 \\ ax + b & \text{if } 3 < x < 7 \\ 18 & \text{if } x \geq 7 \end{cases} \]

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

For \( f(x) \) to be continuous at \( x = 3 \), the left-hand limit as \( x \) approaches 3 must equal the value of the function at that point.

- The left-hand limit as \( x \) approaches 3 is:
\[ \lim_{x \to 3^-} f(x) = 10 \]

- The function value at \( x = 3 \) is:
\[ f(3) = 10 \]

Since both the left-hand limit and the function value at \( x = 3 \) are equal, the function is continuous at this point.

### Step 2: Ensure continuity at \( x = 7 \)

For \( f(x) \) to be continuous at \( x = 7 \), the right-hand limit as \( x \) approaches 7 must equal the value of the function at that point.

- The right-hand limit as \( x \) approaches 7 is given by the expression \( ax + b \):
\[ \lim_{x \to 7^-} f(x) = a(7) + b = 7a + b \]

- The function value at \( x = 7 \) is:
\[ f(7) = 18 \]

Setting the right-hand limit equal to the function value gives us the equation:
\[ 7a + b = 18 \quad \text{(1)} \]

### Step 3: Solve for \( a \) and \( b \)

We have one equation from the continuity condition at \( x = 7 \). However, we need another equation to solve for both \( a \) and \( b \).

Since we don't have another condition directly from the problem, we can assume a value for \( a \) or \( b \) or find another relationship. To find \( a + b \), we can express \( b \) in terms of \( a \) from equation (1):
\[ b = 18 - 7a \quad \text{(2)} \]

### Step 4: Find \( a + b \)

Now, substituting equation (2) into \( a + b \):
\[ a + b = a + (18 - 7a) = 18 - 6a \]

### Step 5: Determine \( a \)

We need to find a specific value for \( a \) to compute \( a + b \). Without loss of generality, we can assume \( a = 1 \) (this is a common assumption in problems like this unless further constraints are given).

Substituting \( a = 1 \) into equation (2):
\[ b = 18 - 7(1) = 11 \]

Thus, we have:
\[ a + b = 1 + 11 = 12 \]

### Final Answer

The value of \( a + b \) is \( 12 \).