For a real number $0 < t < 41$, the curve $y = x ^ { 3 } + 2 x ^ { 2 } - 15 x + 5$ and the line $y = t$ intersect at three points. Let the point with the largest $x$-coordinate be $( f ( t ) , t )$ and the point with the smallest $x$-coordinate be $( g ( t ) , t )$. Let $h ( t ) = t \times \{ f ( t ) - g ( t ) \}$. What is the value of $h ^ { \prime } ( 5 )$? [4 points] (1) $\frac { 79 } { 12 }$ (2) $\frac { 85 } { 12 }$ (3) $\frac { 91 } { 12 }$ (4) $\frac { 97 } { 12 }$ (5) $\frac { 103 } { 12 }$
For a real number $0 < t < 41$, the curve $y = x ^ { 3 } + 2 x ^ { 2 } - 15 x + 5$ and the line $y = t$ intersect at three points. Let the point with the largest $x$-coordinate be $( f ( t ) , t )$ and the point with the smallest $x$-coordinate be $( g ( t ) , t )$. Let $h ( t ) = t \times \{ f ( t ) - g ( t ) \}$. What is the value of $h ^ { \prime } ( 5 )$? [4 points]\\
(1) $\frac { 79 } { 12 }$\\
(2) $\frac { 85 } { 12 }$\\
(3) $\frac { 91 } { 12 }$\\
(4) $\frac { 97 } { 12 }$\\
(5) $\frac { 103 } { 12 }$