$- 8 x ^ { 2 } + 5 x y + y ^ { 3 } = - 149$

a) $- 16 x + 5 y + 5 x a y + 3 y ^ { 2 } d y = 0$

$\operatorname { dy } \left( 5 x + 5 y ^ { 2 } \right) = 16 x - 5 y$

$d x$

$d u = 16 x - 54$

$d x 5 x + 3 y ^ { 2 }$

b) at (4,-1) $\frac { d y } { d x } = \frac { ( 6 \mid 4 ) - 5 ( - 1 ) } { 5 ( 4 ) + 3 ( 1 ) } = \frac { 64 + 5 } { 23 } = \frac { 69 } { 23 } = 3$

Equation of tangent:

$u - ( - 1 ) = 3 ( x - 4 )$

$4 + 1 = 3 x - 12$

$y = 3 x - 13$

C) $( 4,2 , k )$

$k = 3 ( 4.2 ) - 13$

$k = - 0.4$

d) $- 8 ( 4.2 ) ^ { 2 } + 5 ( 4.2 ) ( k ) + k ^ { 3 } = - 149$

$- 141.12 + 21 k + k 3 = - 149$

$k ^ { 3 } + 21 k + 7.88 = 0$

CJUsing G.D.C.

$k = - 0.373$ to 3.d.p