If `y^(2) = ax^(2) + bx + c ` , where a,b,c, are constants , then
` y^(3) (d^(2) y)/(dx^(2))` is equal to

Options

A. constants, then y
B. a constant
C. a function of x
D. a function of y a function of x and y both

Answer

A

Explanation

Given `y^(2)=ax^(2)+bx+c`
On differentiating w.r.t.x. we get
`2y=(dy)/(dx)=2ax+b`
Againg differentiating w.r.t.x. we get
`2((dy)/(dx))^(2)+2y(d^(2)y)/(dx^(2))=2a`
`Rightarrow y(d^(2)y)/(dx^(2))=a-((dy)/(dx))^(2)`
`Rightarrow y(d^(2)y)/(dx^(2))=a((2ax+b)/(2y))^(2)`
`Rightarrow y(d^(2)y)/(dx^(2))=(4ay^(2)-(2ax+b)^(2))/(4y^(2))`
`Rightarrow 4y^(3)(d^(2)y)/(dx^(2))=4a(ax^(2)+bx+c)-4(a^(2)x^(2)+4abx+b^(2))`
`Rightarrow 4y^(3)(d^(2)y)/(dx^2)=4ac-b^(2)`
`Rightarrow y^(3)(d^(2)y)/(dx^2)=(4ac-b^(2))/(4)="constant"`