Question

Problem Set 1 - 1D-6

For each of the following functions, find all values of the constants a and b for which the function is differentiable:

Answer 1

Let take b) as example:

\[ f(x)= \begin{cases}
x^2+4x+1, & x\geq 1 \\
ax+b, & x < 1 \\
\end{cases}
\]

I understand that for a function to be a differentiable one at a certain point, its derivative coming from the left must be the same to that coming from the right. So in this case:

\[\large{\lim_{x\to 0^-}f’(x) = \lim_{x\rightarrow 0^+} f’(x)}\]
\[a= 2x + 4\]
\[a= 2(1) + 4\]
\[a= 6\]
By making the derivative at \(x= 1\) equal when coming from both sides, the function becomes a continuous one as well.

Since the derivative of a constant is zero, does it mean that \(b\) can take any value?

Surely the answer is in the solutions but I don't want to see there yet (: .