Paragraph: Consider the functions defined implicitly by the equation y 3 − 3 y + x = 0 on various intervals in the real line. If x ∈ ( − ∞ , − 2 ) ∪ ( 2 , ∞ ) , the equation implicitly defines a uniqu

Question

Paragraph: Consider the functions defined implicitly by the equation y 3 − 3 y + x = 0 on various intervals in the real line. If x ∈ ( − ∞ , − 2 ) ∪ ( 2 , ∞ ) , the equation implicitly defines a unique real valued differentiable function y = f ( x ) . If x ∈ ( − 2 , 2 ) , the equation implicitly defines a unique real valued differentiable function y = g ( x ) , satisfying g ( 0 ) = 0 . Question: If f ( − 10 2 ​ ) = 2 2 ​ , then f ′′ ( − 10 2 ​ ) is equal to

TestHub · Accepted Answer

Given, y 3 − 3 y + x = 0 ⇒ 3 y 2 d x d y ​ − 3 d x d y ​ + 1 = 0 ⇒ 3 y 2 ( d x 2 d 2 y ​ ) + 6 y ( d x d y ​ ) 2 − 3 d x 2 d 2 y ​ = 0 On substituting x = − 10 2 ​ , y = 2 2 ​ in Eq. (i), we get 3 ( 2 2 ​ ) 2 ⋅ d x d y ​ − 3 ⋅ d x d y ​ + 1 = 0 ⇒ d x d y ​ = 21 − 1 ​ Again on substituting x = − 10 2 ​ , y = 2 2 ​ in Eq. (ii), we get ​ 3 ( 2 2 ​ ) 2 d x 2 d 2 y ​ + 6 ( 2 2 ​ ) ⋅ ( 21 − 1 ​ ) 2 − 3 ⋅ d x 2 d 2 y ​ = 0 ⇒ 21 ⋅ d x 2 d 2 y ​ = − ( 21 ) 2 12 2 ​ ​ ⇒ d x 2 d 2 y ​ = − ( 21 ) 3 12 2 ​ ​ = − 7 3 ⋅ 3 2 4 2 ​ ​ ​

Mathematics - Differentiation Question with Solution | TestHub

Options:

Doubts & Discussion