1. ## piecewise function

$\displaystyle f(x) = \left\{ \begin{array}{lll} x^2 + 2 x + 1 & \mbox{if} & x\le -2 \cr ax + b & \mbox{if} & x>-2 \end{array} \right .$

This is differentiable at -2 .

(a) It is known that if a function is differentiable at a point c, then it is continuous at c. Using now the continuity of f at −2, we can establish a relationship between a and b. Find this relationship and express it in the form b=Aa+B, where A and B are constants.

b) Assuming x>-2

$\displaystyle \displaystyle \frac{f(x)-f(-2)}{x-(-2)}$

simplify into the form Ca+D, where C and D are constants. Find these constants.

c) assuming x<-2

$\displaystyle \displaystyle \frac{f(x)-f(-2)}{x-(-2)}$

simplify into the form Ex+F, where E and F are constants. Find these constants.

(d) Using the results of parts (a), (b) and (c), find the values of a and b.

This is the part I don't understand how to do. The answer is allegedly -2 & -3.

Can someone please explain how to do this part "D"?

Thank you!

2. Continuity gives $\displaystyle -2a+b=1$.

Differentiability gives $\displaystyle a=-2$

3. Got it, Thanks.