Let f ( x ) = x ^ 3 + x ^ 2 f ^ ' ( 1 ) + x f ^ " ( 2 ) + f ^ ''' (...

Given: $f ( x ) = {x}^{3} + {x}^{2}{f}^{'} ( 1 ) + x{f}^{"} ( 2 ) + {f}^{'''} ( 3 )$

$⇒ {f}^{'} ( x ) = 3{x}^{2} + 2x{f}^{'} ( 1 ) + {f}^{"} ( 2 ) . . . \left(i\right)</p><p>$⇒ {f}^{"} ( x ) = 6x + 2{f}^{'} ( 1 ) . . . \left(ii\right)$</p><p>$⇒ {f}^{'''} ( x ) = 6$</p><p>$⇒ {f}^{'''} ( 3 ) = 6 . . . \left(iii\right)

Using \left(ii\right),

$⇒ {f}^{"} ( 2 ) = 12 + 2{f}^{'} ( 1 )</p><p>Putting this value in$equation \left(i\right),

$⇒ {f}^{'} ( 1 ) = 3 + 2{f}^{'} ( 1 ) + 12 + 2{f}^{'} ( 1 )$

$⇒ 0 = 15 + 3{f}^{'} ( 1 )$

$⇒ {f}^{'} ( 1 ) = - 5 . . . \left(iv\right)$

$⇒ {f}^{"} ( 2 ) = 12 + 2\left(- 5\right)</p><p>$⇒ {f}^{"} ( 2 ) = 2 . . . \left(v\right)$</p><p>$⇒ f ( x ) = {x}^{3} + {x}^{2} \cdot ( - 5 ) + x \cdot ( 2 ) + 6$</p><p>$⇒ {f}^{'} ( x ) = 3{x}^{2} - 10x + 2$</p><p>$⇒ {f}^{'} ( 10 ) = 300 - 100 + 2$</p><p>$⇒ {f}^{'} ( 10 ) = 202$