Question

If $\left(\sin x\right)\left(\cos y\right)=\frac{1}{2}$ then $\frac{d^{2}y}{d{x}^2}$ at $(\frac{\pi }{4},\frac{\pi }{4})$ is equal to

• A. $-4$
• B. $-2$
• C. $-6$
• D. $0$

Answer 1

The correct option is A $-4$

Differentiating both sides, we have

$\cos x\cos y-\sin x\sin y\frac{dy}{dx}=0$

Putting $x=y=\frac{\pi }{4}$

We have $\frac{dy}{dx}{|}_{(\frac{\pi }{4},\frac{\pi }{4})}=1$

Differentiating again, we get

$-\sin x\cos y-\cos x\sin y\left(\frac{dy}{dx}\right)-\cos x\sin y\frac{dy}{dx}$

$-\sin x\cos y{\left(\frac{dy}{dx}\right)}^2-\sin x\sin y\frac{d^{2}y}{{dx}^2}=0$

Putting $x=y=\frac{\pi }{4}$

$\frac{d^{2}y}{d{x}^2}{|}_{(\frac{\pi }{4},\frac{\pi }{4})}=-4$