$$ s(x)=\left\{\begin{array}{ll} x^{2}+1, & x<0 \\ x+2, & 0 \leq x<2 \\ x^{2}, & x \geq 2 \end{array}\right. $$
a) In which points is the function derivable
b) If possible calculate $s'(1)$ and give the equation for the tangent for $y= s(x)$ when $x =1$
The attempt at the questions
a) Check if the three functions are derivable by computing $\lim x$ go towards $0$ and $2$. The functions do not get same answer, thus the function $s(x)$ is not derivable in the points $x=0$ and $x=2$ ?
a) you are correct here
b) As Emilio Novati stated a couple seconds ago the derivative is piecewise: $$s'(x)=\begin{cases} 2x \qquad x<0\\ 1\qquad 0<x<2\\ 2x \qquad x>2 \end{cases}$$ Which is true. However, because $x=1$ falls into the 2nd condition, which is linear, and the tangent line of a line is just a line, at $s(1)$ the tangent line is $x+2$