Question

If \(y = \sqrt{x^2+8x+3}\), then the value of \(\frac{d^2y}{dx^2}\) at \(x = 3\) is

• A. \(\frac{-13}{216}\)
• B. \(\frac{13}{216}\)
• C. \(\frac{15}{216}\)
• D. \(\frac{-15}{216}\)

Hint:

Use $y^2=x^2+8x+3$ and differentiate twice.

Answer 1

The Correct Option is A

Step 1: Write $y=u^{1/2}$
With $u=x^2+8x+3$, $y'=\frac{u'}{2\sqrt u}$ and $u'=2x+8$, $u''=2$.
$y''=\frac{u''}{2\sqrt u}-\frac{(u')^2}{4u^{3/2}}$.

Step 2: Plug in $x=3$
$u=36$, $u'=14$, $u''=2$: $y''=\frac{2}{12}-\frac{196}{4\times216}=\frac{1}{6}-\frac{49}{216}=\frac{36-49}{216}=-\frac{13}{216}$.

Final Answer:
Option (A). \[ \boxed{\text{(A)}} \]