Problem
Find all values of $a$ and $b$ that make the following function differentiable for all values of $x$:
$$
f(x) =
\begin{cases}
ax + b, x > – 1\\
bx^2 – 3, x \leq -1\\
\end{cases}
$$
I was reviewing all my works for the last 3 years, and suddenly I found this problem in one of my homework assignment. Unfortunately, this assignment lacks the solution. I was 100% sure that I used to solve it correctly, but without a textbook I couldn't recall my poor memory. I knew this is a very basic question for the first calculus course, but I totally forgot how to start, could anyone give me a "tiny" hint? Thank you.
Best Answer
The only value of $x$ that gives any trouble is $x=-1$. You want $f$ to be continuous there, which gives you one equation in $a$ and $b$, and you want it to have the same derivative whichever formula you use, and that gives you a second equation.