1. ## Equalling Derivatives

Suppose the function g is defined by
g(x) = k$\displaystyle sqrt(x + 1)$ for 0 $\displaystyle \le$ x $\displaystyle \le$ 3

= mx + 2 for 3 < x $\displaystyle \le$ 5

where k and m are constants. If g is differentiable at x = 3, what are the values of k and m.
this is part c on a question, but i dont think you need the information from the original question.
Thanks!

2. If it is given that $\displaystyle g(x)$ is differentiable at $\displaystyle x=3$, then both pieces of the piecewise function must have the same values of $\displaystyle g(3)$ and $\displaystyle g'(3)$.

So let's see where the two pieces are equal to each other when $\displaystyle x=3$:

$\displaystyle k\sqrt{3+1}=3m+2$
$\displaystyle k=\frac{3m+2}{2}$

Making that substitution, let's see where the derivatives are equal to each other at $\displaystyle x=3$:

$\displaystyle \frac{\frac{3m+2}{2}}{2\sqrt{3+1}}=m$
$\displaystyle \frac{3m+2}{8}=m$
$\displaystyle m=\frac{2}{5}$
$\displaystyle k=\frac{3(\frac{2}{5})+2}{2}=\frac{8}{5}$

So, $\displaystyle g(x)$ is differentiable at $\displaystyle k=\frac{8}{5},m=\frac{2}{5}$. You can check your work to see that this piecewise function is continuous at $\displaystyle x=3$ with these constants and that there is a single value for $\displaystyle g'(3)$ with these constants.

3. Originally Posted by Pinkk
If it is given that $\displaystyle g(x)$ is differentiable at $\displaystyle x=3$, then both pieces of the piecewise function must have the same values of $\displaystyle g(3)$ and $\displaystyle g'(3)$.

So let's see where the two pieces are equal to each other when $\displaystyle x=3$:

$\displaystyle k\sqrt{3+1}=3m+2$
$\displaystyle k=\frac{3m+2}{2}$

Making that substitution, let's see where the derivatives are equal to each other at $\displaystyle x=3$:

$\displaystyle \frac{\frac{3m+2}{2}}{2\sqrt{3+1}}=m$
$\displaystyle \frac{3m+2}{8}=m$
$\displaystyle m=\frac{2}{5}$
$\displaystyle k=\frac{3(\frac{2}{5})+2}{2}=\frac{8}{5}$

So, $\displaystyle g(x)$ is differentiable at $\displaystyle k=\frac{8}{5},m=\frac{2}{5}$. You can check your work to see that this piecewise function is continuous at $\displaystyle x=3$ with these constants and that there is a single value for $\displaystyle g'(3)$ with these constants.
thanks so much. its so simple once u explain it