7. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 4,25 )\) Given that $$\mathrm { f } ^ { \prime } ( x ) = \frac { 3 } { 8 } x ^ { 2 } - 10 x ^ { - \frac { 1 } { 2 } } + 1 , \quad x > 0$$ find \(\mathrm { f } ( x )\), simplifying each term. $$\begin{aligned} & \therefore F ( x ) = \int F ^ { \prime } ( x ) = \int 3 / 8 x ^ { 2 } - 10 x ^ { - 1 / 2 } + 1 d x
& F ( x ) = \frac { 3 x ^ { 3 } } { 8 ( 3 ) } - \frac { 10 x ^ { 1 / 2 } } { 1 / 2 } + x + c
& F ( x ) = 1 / 8 x ^ { 3 } - 20 x ^ { 1 / 2 } + x + c
& 25 = 1 / 8 ( 4 ) ^ { 3 } - 20 ( 4 ) ^ { 1 / 2 } + 4 + c
& 25 = 8 - 40 + 4 + c
& C = 53
& F ( x ) = 1 / 8 x ^ { 3 } - 20 x ^ { 1 / 2 } + x + 53 \end{aligned}$$ \section*{PMT PhysicsAndMathsTutor.com}