Questions: For the function f(x) = x^2 / (2 + x), find f''(x). Then find f''(0) and f''(4).

Transcript text: For the function $f(x)=\frac{x^{2}}{2+x}$, find $f^{\prime \prime}(x)$. Then find $f^{\prime \prime}(0)$ and $f^{\prime \prime}(4)$.

Solution

Step 1: Find the first derivative, $f'(x)$

Using the quotient rule, $f'(x) = \frac{d}{dx}\left(\frac{x^2}{n + x}\right)$ results in: $f'(x) = -x^2/(x + 2)^2 + 2*x/(x + 2)$

Step 2: Find the second derivative, $f''(x)$

Differentiating $f'(x)$ again, we get: $f''(x) = 2_x^2/(x + 2)^3 - 4_x/(x + 2)^2 + 2/(x + 2)$

Step 3: Evaluate $f''(x)$ at specific points

At $x = 0$, $f''(0) = 1$ At $x = 4$, $f''(4) = 0.037$

$f''(0) = 1$ $f''(4) = 0.037$

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