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 $\mathbf { U } < \mathbf { V }$.
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 $\mathbf { U } < \mathbf { V }$.