$\text { Find the cubic function of the form } f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d \text {, where } a \neq 0 \text { and the }$

Multiple Choice

$\text { Find the cubic function of the form } f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d \text {, where } a \neq 0 \text { and the }$ coefficients $a , b , c , d$ are real numbers, which satisfies the conditions given below.
Relative maximum: $( 3,0 )$
Relative minimum: $( 5 , - 2 )$
Inflection point: $( 4 , - 1 )$

A) $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 6 x ^ { 2 } + \frac { 45 } { 2 } x - 27$
B) $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 6 x ^ { 2 } + \frac { 41 } { 2 } x - 26$
C) $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 6 x ^ { 2 } + \frac { 45 } { 2 } x + 27$
D) $f ( x ) = - \frac { 1 } { 2 } x ^ { 3 } + 6 x ^ { 2 } - \frac { 45 } { 2 } x + 27$
E) $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 6 x ^ { 2 } + \frac { 41 } { 2 } x + 26$