if the function `g(x)={{:(ksqrt(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. 2
B. `16/5`
C. `10/3`
D. 4

Answer

To solve the problem, we need to ensure that the piecewise function \( g(x) \) is both continuous and differentiable at the point \( x = 3 \). The function is defined as follows:

\[ g(x) = \begin{cases} k \sqrt{x + 1} & \text{for } 0 \leq x \leq 3 \\ mx + 2 & \text{for } 3 < x \leq 5 \end{cases} \]

### 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 \( g(3) \).

1. **Left-hand limit at \( x = 3 \)**:
\[ g(3^-) = k \sqrt{3 + 1} = k \sqrt{4} = 2k \]

2. **Right-hand limit at \( x = 3 \)**:
\[ g(3^+) = m(3) + 2 = 3m + 2 \]

Setting these equal for continuity:
\[ 2k = 3m + 2 \tag{1} \]

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

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

1. **Derivative from the left**:
\[ g'(x) = \frac{d}{dx}(k \sqrt{x + 1}) = \frac{k}{2\sqrt{x + 1}} \]
Evaluating at \( x = 3 \):
\[ g'(3^-) = \frac{k}{2\sqrt{4}} = \frac{k}{4} \]

2. **Derivative from the right**:
\[ g'(x) = \frac{d}{dx}(mx + 2) = m \]
Evaluating at \( x = 3 \):
\[ g'(3^+) = m \]

Setting these equal for differentiability:
\[ \frac{k}{4} = m \tag{2} \]

### Step 3: Solve the System of Equations

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

Substituting Equation (2) into Equation (1):
\[ 2k = 3\left(\frac{k}{4}\right) + 2 \]
Multiplying through by 4 to eliminate the fraction:
\[ 8k = 3k + 8 \]
Rearranging gives:
\[ 8k - 3k = 8 \implies 5k = 8 \implies k = \frac{8}{5} \]

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

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

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

### Final Answer
Thus, the value of \( k + m \) is \( \boxed{2} \).