If the function `f(x)={{:(asqrt(x+7),":",0lexlt9),(bx+5,":",xge9):}` is differentiable for `xge0`, then the value of `5a+6b` is equal to

Options

A. `(240)/(23)`
B. 10
C. `(80)/(23)`
D. `(250)/(23)`

Answer

To solve the problem, we need to ensure that the function \( f(x) \) is both continuous and differentiable at the point where the definition of the function changes, which is at \( x = 9 \).

### Step 1: Define the function
The function is defined as:
\[ f(x) = \begin{cases} a \sqrt{x + 7} & \text{for } 0 \leq x < 9 \\ bx + 5 & \text{for } x \geq 9 \end{cases} \]

### Step 2: Ensure continuity at \( x = 9 \)
For \( f(x) \) to be continuous at \( x = 9 \), we must have:
\[ f(9^-) = f(9^+ \]
Calculating \( f(9^-) \):
\[ f(9^-) = a \sqrt{9 + 7} = a \sqrt{16} = 4a \]
Calculating \( f(9^+) \):
\[ f(9^+) = b(9) + 5 = 9b + 5 \]
Setting these equal for continuity:
\[ 4a = 9b + 5 \quad \text{(Equation 1)} \]

### Step 3: Ensure differentiability at \( x = 9 \)
For \( f(x) \) to be differentiable at \( x = 9 \), we must have:
\[ f'(9^-) = f'(9^+) \]
Calculating \( f'(9^-) \):
\[ f'(x) = \frac{a}{2\sqrt{x + 7}} \quad \text{for } 0 \leq x < 9 \]
So,
\[ f'(9^-) = \frac{a}{2\sqrt{16}} = \frac{a}{8} \]
Calculating \( f'(9^+) \):
\[ f'(x) = b \quad \text{for } x \geq 9 \]
So,
\[ f'(9^+) = b \]
Setting these equal for differentiability:
\[ \frac{a}{8} = b \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 2 into Equation 1
From Equation 2, we can express \( a \) in terms of \( b \):
\[ a = 8b \]
Substituting into Equation 1:
\[ 4(8b) = 9b + 5 \]
This simplifies to:
\[ 32b = 9b + 5 \]
\[ 32b - 9b = 5 \]
\[ 23b = 5 \]
\[ b = \frac{5}{23} \]

### Step 5: Find \( a \)
Using \( b \) to find \( a \):
\[ a = 8b = 8 \left(\frac{5}{23}\right) = \frac{40}{23} \]

### Step 6: Calculate \( 5a + 6b \)
Now we compute \( 5a + 6b \):
\[ 5a = 5 \left(\frac{40}{23}\right) = \frac{200}{23} \]
\[ 6b = 6 \left(\frac{5}{23}\right) = \frac{30}{23} \]
Adding these:
\[ 5a + 6b = \frac{200}{23} + \frac{30}{23} = \frac{230}{23} = 10 \]

### Final Answer
Thus, the value of \( 5a + 6b \) is \( \boxed{10} \).