Question

If $x^2+y^2+\sin y=4,$ then the value of $\frac{d^{2}y}{d{x}^2}$ at the point $(-2,0)$ is :

• A. $-34$
• B. $-32$
• C. $4$
• D. $-2$

Answer 1

The correct option is A $-34$

$\begin{array}{c}Wehave\\ x^2+y^2+\sin y=4\\ =2x+2y\frac{dy}{dx}+\cos y\frac{dy}{dx}=0\\ =2+2\left[{\left(\frac{dy}{dx}\right)}^2+y\frac{d^{2}y}{d{x}^2}\right]+\cos \frac{d^{2}y}{dx}-{\left(\frac{dy}{dx}\right)}^2\sin y=0\\ Now,\\ At\left(-2,0\right)\\ -4+\frac{dy}{dx}=0\\ \frac{dy}{dx}=4\\ Now,\\ =2+2\left[{\left(\frac{dy}{dx}\right)}^2+y\frac{d^{2}y}{d{x}^2}\right]+\cos \frac{d^{2}y}{dx}-{\left(\frac{dy}{dx}\right)}^2\sin y=0\\ At\left(-2,0\right)\\ 2+2\left[{\left(\frac{dy}{dx}\right)}^2\right]+\frac{d^{2}y}{d{x}^2}=0\\ \therefore \frac{d^{2}y}{d{x}^2}=-34\\ hence,theoptionAisthecorrectanswer.\end{array}$