If the function `g (x ) = {{:( k sqrt (x + 1)",", 0 le x le 3),( mx + 2 "," , 3 lt x le 5):}` is differentiable, then the value of `k + m` is :

Options

A. `(10)/(3)`
B. `4`
C. 2
D. `(16)/(5)`

Explanation

To solve the problem, we need to ensure that the function \( g(x) \) is continuous and differentiable at \( x = 3 \). The function is defined in two pieces:

1. \( g(x) = k \sqrt{x + 1} \) for \( 0 \leq x \leq 3 \)
2. \( g(x) = mx + 2 \) for \( 3 < x \leq 5 \)

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

For \( g(x) \) to be continuous at \( x = 3 \), the left-hand limit as \( x \) approaches 3 from the left must equal the right-hand limit as \( x \) approaches 3 from the right, and both must equal \( g(3) \).

Calculating the left-hand limit:
\[ g(3) = k \sqrt{3 + 1} = k \cdot 2 = 2k \]

Calculating the right-hand limit:
\[ g(3) = m \cdot 3 + 2 = 3m + 2 \]

Setting these equal for continuity:
\[ 2k = 3m + 2 \quad \text{(1)} \]

### Step 2: Ensure Differentiability at \( x = 3 \)

For \( g(x) \) to be differentiable at \( x = 3 \), the left-hand derivative must equal the right-hand derivative.

Calculating the left-hand derivative:
\[ g'(x) = \frac{d}{dx}(k \sqrt{x + 1}) = \frac{k}{2\sqrt{x + 1}} \]
At \( x = 3 \):
\[ g'(3) = \frac{k}{2\sqrt{4}} = \frac{k}{4} \]

Calculating the right-hand derivative:
\[ g'(x) = m \quad \text{(since the derivative of \( mx + 2 \) is \( m \))} \]
At \( x = 3 \):
\[ g'(3) = m \]

Setting these equal for differentiability:
\[ \frac{k}{4} = m \quad \text{(2)} \]

### Step 3: Solve the Equations

Now we have two equations:
1. \( 2k = 3m + 2 \)
2. \( m = \frac{k}{4} \)

Substituting equation (2) into equation (1):
\[ 2k = 3\left(\frac{k}{4}\right) + 2 \]
\[ 2k = \frac{3k}{4} + 2 \]

Multiplying through by 4 to eliminate the fraction:
\[ 8k = 3k + 8 \]
\[ 8k - 3k = 8 \]
\[ 5k = 8 \quad \Rightarrow \quad k = \frac{8}{5} \]

Now substituting \( k \) back into equation (2) to find \( m \):
\[ m = \frac{8/5}{4} = \frac{8}{20} = \frac{2}{5} \]

### Step 4: Find \( k + m \)

Now we can find \( k + m \):
\[ k + m = \frac{8}{5} + \frac{2}{5} = \frac{10}{5} = 2 \]

Thus, the final answer is:
\[ \boxed{2} \]