Consider the function $$f ( x ) = \left\{ \begin{array} { l l l }
\operatorname { sen } x & \text { if } & x < 0 \\
x e ^ { x } & \text { if } & x \geq 0
\end{array} \right.$$ a) ( 0.75 points) Study the continuity and differentiability of $f$ at $x = 0$.\ b) (1 point) Study the intervals of increase and decrease of $f$ restricted to ( $- \pi , 2$ ). Prove that there exists a point $x _ { 0 } \in [ 0,1 ]$ such that $f \left( x _ { 0 } \right) = 2$.\ c) (0.75 points) Calculate $\int _ { - \frac { \pi } { 2 } } ^ { 1 } f ( x ) d x$.
Consider the function
$$f ( x ) = \left\{ \begin{array} { l l l }
\operatorname { sen } x & \text { if } & x < 0 \\
x e ^ { x } & \text { if } & x \geq 0
\end{array} \right.$$
a) ( 0.75 points) Study the continuity and differentiability of $f$ at $x = 0$.\
b) (1 point) Study the intervals of increase and decrease of $f$ restricted to ( $- \pi , 2$ ). Prove that there exists a point $x _ { 0 } \in [ 0,1 ]$ such that $f \left( x _ { 0 } \right) = 2$.\
c) (0.75 points) Calculate $\int _ { - \frac { \pi } { 2 } } ^ { 1 } f ( x ) d x$.