Let $f$ be a function that is differentiable for all real numbers. The table below gives the values of $f$ and its derivative $f ^ { \prime }$ for selected points $x$ in the closed interval $- 1.5 \leq x \leq 1.5$. The second derivative of $f$ has the property that $f ^ { \prime \prime } ( x ) > 0$ for $- 1.5 \leq x \leq 1.5$.
| $x$ | - 1.5 | - 1.0 | - 0.5 | 0 | 0.5 | 1.0 | 1.5 |
| $f ( x )$ | - 1 | - 4 | - 6 | - 7 | - 6 | - 4 | - 1 |
| $f ^ { \prime } ( x )$ | - 7 | - 5 | - 3 | 0 | 3 | 5 | 7 |
(a) Evaluate $\int _ { 0 } ^ { 1.5 } \left( 3 f ^ { \prime } ( x ) + 4 \right) d x$. Show the work that leads to your answer.
(b) Write an equation of the line tangent to the graph of $f$ at the point where $x = 1$. Use this line to approximate the value of $f ( 1.2 )$. Is this approximation greater than or less than the actual value of $f ( 1.2 )$? Give a reason for your answer.
(c) Find a positive real number $r$ having the property that there must exist a value $c$ with $0 < c < 0.5$ and $f ^ { \prime \prime } ( c ) = r$. Give a reason for your answer.
(d) Let $g$ be the function given by $g ( x ) = \begin{cases} 2 x ^ { 2 } - x - 7 & \text { for } x < 0 \\ 2 x ^ { 2 } + x - 7 & \text { for } x \geq 0 . \end{cases}$ The graph of $g$ passes through each of the points $( x , f ( x ) )$ given in the table above. Is it possible that $f$ and $g$ are the same function? Give a reason for your answer.