Question

If $u = x^2 + y^2$ and $x = s + 3t$, $y = 2s - t$, then $\frac{d^{2}u}{ds^{2}}$ is equal to

• A. $12$
• B. $32$
• C. $36$
• D. $10$

Answer 1

Answer: (D)

Given, $u = x^2+y^2$,
$x = s + 3t$,
$y = 2s - t$
$\Rightarrow \frac{dx}{ds}=1, \frac{dy}{ds}=2$
Now, $u=x^{2}+y^{2}$
$\Rightarrow \frac{du}{ds}=2x \frac{dx}{ds}+2y \frac{dy}{ds}$
$=2x+4y$
$\frac{d^{2}u}{ds^{2}}=2\left(\frac{dx}{ds}\right)+4\left(\frac{dy}{ds}\right)$
$\Rightarrow \frac{d^{2}u}{ds^{2}}=2\left(1\right)+4\left(2\right)$
$=10$