Question:

Step 1: Define the condition for continuity at $x=3$. For the function $g(x)$ to be continuous at $x=3$, the left-hand limit, the right-hand limit, and the function value at $x=3$ must all be equal. That is, $\lim_{x \to 3^-} g(x) = \lim_{x \to 3^+} g(x) = g(3)$. Step 2: Calculate the left-hand limit of $g(x)$ as $x \to 3^-$. For $0 \leq x \leq 3$, $g(x) = k\sqrt{x+1}$. $$ \lim_{x \to 3^-} g(x) = \lim_{x \to 3^-} k\sqrt{x+1} = k\sqrt{3+1} = k\sqrt{4} = 2k $$ Step 3: Calculate the right-hand limit of $g(x)$ as $x \to 3^+$. For $x > 3$, $g(x) = mx+2$. $$ \lim_{x \to 3^+} g(x) = \lim_{x \to 3^+} (mx+2) = m(3)+2 = 3m+2 $$ The function value at $x=3$ is $g(3) = k\sqrt{3+1} = 2k$. Step 4: Equate the limits to establish the first equation for $k$ and $m$. From the continuity condition, we set the left-hand limit equal to the right-hand limit: $$ 2k = 3m+2 \quad (*)$$ Step 5: Define the condition for differentiability at $x=3$. For the function $g(x)$ to be differentiable at $x=3$, it must first be continuous at $x=3$, and its left-hand derivative must be equal to its right-hand derivative at $x=3$. That is, $g'(3^-) = g'(3^+)$. Step 6: Calculate the left-hand derivative of $g(x)$ at $x=3$. For $x < 3$, $g(x) = k\sqrt{x+1}$. We find its derivative: $$ g'(x) = \frac{d}{dx} (k(x+1)^{1/2}) = k \cdot \frac{1}{2}(x+1)^{-1/2} = \frac{k}{2\sqrt{x+1}} $$ Now, evaluate the left-hand derivative at $x=3$: $$ g'(3^-) = \frac{k}{2\sqrt{3+1}} = \frac{k}{2\sqrt{4}} = \frac{k}{2 \cdot 2} = \frac{k}{4} $$ Step 7: Calculate the right-hand derivative of $g(x)$ at $x=3$. For $x > 3$, $g(x) = mx+2$. We find its derivative: $$ g'(x) = \frac{d}{dx} (mx+2) = m $$ Now, evaluate the right-hand derivative at $x=3$: $$ g'(3^+) = m $$ Step 8: Equate the derivatives to establish the second equation for $k$ and $m$. From the differentiability condition, we set the left-hand derivative equal to the right-hand derivative: $$ \frac{k}{4} = m $$ This gives us a relationship between $k$ and $m$: $$ k = 4m \quad (**)$$ Step 9: Solve the system of equations for $m$. Substitute the expression for $k$ from equation $(**)$ into equation $(*)$: $$ 2k = 3m+2 $$ $$ 2(4m) = 3m+2 $$ $$ 8m = 3m+2 $$ $$ 8m - 3m = 2 $$ $$ 5m = 2 $$ $$ m = \frac{2}{5} $$ Step 10: Substitute the value of $m$ to find the value of $k$. Using the relationship $k = 4m$: $$ k = 4 \left(\frac{2}{5}\right) $$ $$ k = \frac{8}{5} $$ Step 11: State the final answer. The calculated value for $k$ is $\frac{8}{5}$. The original solution indicates Option D as the correct answer.