1. ## Differentiation

I need some help with this problem (please don't give me the answer though, just hint me)

suppose the function g is defined by

k sqrt(X+1) for 0<(including) x<(including) 3
g(X)=<
mX+2 for 3<x<(including) 5

where k and m are constants. if g is differentiable at x=3, what are the values of k and m.

Im not sure how to do this. But to start I tried solving the the two pieces of the function and got
g(X)= 2k & g(X)=3m+2

2. Hello, rawkstar!

$g(x) \;=\;\begin{Bmatrix}k\sqrt{x+1} & 0 \leq x\leq 3 \\
mx+2 & 3 < x \leq 5 \end{Bmatrix}$
.where $k$ and $m$ are constants.

If $g(x)$ is differentiable at $x=3$, what are the values of $k$ and $m$ ?

Since the function is differentiable at $x=3$
. . the two branches of the function are equal when $x = 3.$

We have: . $k\sqrt{3+1} \:=\:m(3) + 2\quad\Rightarrow\quad\boxed{ 2k \:=\:3m + 2}$

The derivatives of the two branches are: . $g'(x) \;=\;\begin{Bmatrix}\dfrac{k}{2\sqrt{x+1}} \\ m \end{Bmatrix}$

Since the derivatives are equal at $x = 3$, we have:

. . $\frac{k}{2\sqrt{3+1}} \:=\:m \quad\Rightarrow\quad\boxed{ k \:=\:4m}$

Now solve the system of equations . . .

3. how did you get the derivative and why do you know that?