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!

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

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

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

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

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

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

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

Now solve the system of equations . . .

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