If x2+y2+siny=4 then d2ydx2 at -2,0 is

x2+y2+sin y=4 2x+2yy1+cos yy1=0 →(1) 2+2yy11+2(y1)2-sin yy1+cos yy11=0 →(2) If x=-2, y=0 then (1) ⇒ -4+0+y1=0 y1=4 now (2) ⇒ 2+0+2(4)2-0+y11=0 y11=-34

If x2+y2+siny=4 then d2ydx2 at -2,0 is

x2+y2+sin y=4 2x+2yy1+cos yy1=0 →(1) 2+2yy11+2(y1)2-sin yy1+cos yy11=0 →(2) If x=-2, y=0 then (1) ⇒ -4+0+y1=0 y1=4 now (2) ⇒ 2+0+2(4)2-0+y11=0 y11=-34

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$If x^{2} + y^{2} + siny = 4 then \frac{d^{2} y}{{dx}^{2}} at (- 2, 0) is$

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$- 32$

$4$

$- 2$

answer is A.

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Detailed Solution

$x^{2} + y^{2} + \sin y = 4 2 x + 2 {yy}^{1} + \cos {yy}^{1} = 0 \to (1) 2 + 2 {yy}^{11} + 2 {(y^{1})}^{2} - \sin {yy}^{1} + \cos {yy}^{11} = 0 \to (2) If x = - 2, y = 0 then (1) \Rightarrow - 4 + 0 + y^{1} = 0 y^{1} = 4 now (2) \Rightarrow 2 + 0 + 2 {(4)}^{2} - 0 + y^{11} = 0 y^{11} = - 34$