Determine all real values of a and b such that the function
f(x) = ax+b for x =< 1
= tan(Pi*x/4) for 1 < x < 2
is differentiable at x = 1
?
Is this the correct function?
$\displaystyle f(x) = \begin{cases} ax+b, & \mbox{if } x \le 1 \\ \tan ( \pi x/4 ), & \mbox{if } 1<x<2 \end{cases}$
You basically need to check for two things:
1) The function must be equal on both sides as $\displaystyle x \to 1$
2) The function's derivative must be equal on both sides as $\displaystyle x \to 1$
For the first condition, plug in $\displaystyle x=1$ into both parts to get:
$\displaystyle a+b = \tan(\pi / 4)$
And the second condition, take derivatives of both sides, then plug in $\displaystyle x=1$ to get:
$\displaystyle a = \frac{\pi}{4} \sec^2 (\pi / 4)$
Once you have established these two equalities, you can now determine what $\displaystyle a,b$ are.