Given a continuous function $f$, define the following subsets of the set $\mathbb{R}$ of real numbers.
$T =$ set of slopes of all possible tangents to the graph of $f$.
$S =$ set of slopes of all possible secants, i.e. lines joining two points on the graph of $f$.
For each statement below, state if it is true or false.
(i) If $f$ is differentiable, then $S \subset T$.
(ii) If $f$ is differentiable, then $T \subset S$.
(iii) If $T = S = \mathbb{R}$, then $f$ must be differentiable everywhere.
(iv) Suppose 0 and 1 are in $S$. Then every number between 0 and 1 must also be in $S$.

15. If the curve $y = ( x + a ) \mathrm { e } ^ { x }$ has two tangent lines passing through the origin, then the range of $a$ is $\_\_\_\_$ .

Let $f ( x ) = x ^ { 4 } + 2 x ^ { 3 } - 12 x ^ { 2 } + 4$. We are to find the values of $p$ such that we can draw three tangents to the curve $y = f ( x )$ from the point $\mathrm { P } ( 0 , p )$ on the $y$-axis.
(i) The equation of the tangent to the curve $y = f ( x )$ at the point $( t , f ( t ) )$ is
$$y = \left( \mathbf { A } t ^ { 3 } + \mathbf { B } t ^ { 2 } - \mathbf { C D } t \right) x - \mathbf { E } t ^ { 4 } - \mathbf { F } t ^ { 3 } + \mathbf { G H } t ^ { 2 } + \mathbf { I }$$
The condition under which this straight line passes through the point $\mathrm { P } ( 0 , p )$ is that
$$p = - \mathbf { J } t ^ { 4 } - \mathbf { K } t ^ { 3 } + \mathbf { L M } t ^ { 2 } + \mathbf { N }$$
(ii) For $\mathbf { O }$ and $\mathbf { S }$ in the following statements, choose either (0) or (1) and for the other blanks, enter the correct number. (0) local minimum
(1) local maximum
When the right side of (1) is set to $g ( t )$, the function $g ( t )$ takes a $\mathbf{O}$ at $t = \mathbf { P Q }$ and $t = \mathbf { R }$. On the other hand, $g ( t )$ takes a $\mathbf { S }$ at $t = \mathbf { T }$.
Hence the values of $p$ such that we can draw three tangents to the curve $y = f ( x )$ from the point $\mathrm { P } ( 0 , p )$ are
$$p = \mathbf { U } \text { and } p = \mathbf { V } ,$$
where $\mathbf { U } < \mathbf { V }$.