for the function $f(x)=\\frac{x^{2}}{5 + x}$, find $f(x)$. then find $f(0)$ and $f(2)$.\n$f(x)=\\square$\nsel…

for the function $f(x)=\\frac{x^{2}}{5 + x}$, find $f(x)$. then find $f(0)$ and $f(2)$.\n$f(x)=\\square$\nselect the correct choice below and fill in any answer boxes in your choice.\na. $f(0)=\\square$ (simplify your answer. type an exact answer.)\nb. $f(0)$ is undefined.\nselect the correct choice below and fill in any answer boxes in your choice.\na. $f(2)=\\square$ (simplify your answer. type an exact answer.)\nb. $f(2)$ is undefined.

for the function $f(x)=\\frac{x^{2}}{5 + x}$, find $f(x)$. then find $f(0)$ and $f(2)$.\n$f(x)=\\square$\nselect the correct choice below and fill in any answer boxes in your choice.\na. $f(0)=\\square$ (simplify your answer. type an exact answer.)\nb. $f(0)$ is undefined.\nselect the correct choice below and fill in any answer boxes in your choice.\na. $f(2)=\\square$ (simplify your answer. type an exact answer.)\nb. $f(2)$ is undefined.

Answer

Explanation:

Step1: Find the first - derivative

Use the power rule (y = ax^n), (y^\prime=anx^{n - 1}). Given (f(x)=\frac{x^{2}}{5 + x}), by the quotient rule ((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}), where (u = x^{2}), (u^\prime=2x), (v = 5 + x), (v^\prime = 1). (f^\prime(x)=\frac{2x(5 + x)-x^{2}\times1}{(5 + x)^{2}}=\frac{10x+2x^{2}-x^{2}}{(5 + x)^{2}}=\frac{x^{2}+10x}{(5 + x)^{2}}).

Step2: Find the second - derivative

Again use the quotient rule. Let (u=x^{2}+10x), (u^\prime = 2x + 10), (v=(5 + x)^{2}), (v^\prime=2(5 + x)). (f^{\prime\prime}(x)=\frac{(2x + 10)(5 + x)^{2}-(x^{2}+10x)\times2(5 + x)}{(5 + x)^{4}}). Factor out ((5 + x)) from the numerator: (f^{\prime\prime}(x)=\frac{(5 + x)[(2x + 10)(5 + x)-2(x^{2}+10x)]}{(5 + x)^{4}}=\frac{(2x + 10)(5 + x)-2(x^{2}+10x)}{(5 + x)^{3}}). Expand ((2x + 10)(5 + x)=10x+2x^{2}+50 + 10x=2x^{2}+20x + 50). (f^{\prime\prime}(x)=\frac{2x^{2}+20x + 50-2x^{2}-20x}{(5 + x)^{3}}=\frac{50}{(5 + x)^{3}}).

Step3: Evaluate (f^{\prime\prime}(0))

Substitute (x = 0) into (f^{\prime\prime}(x)): (f^{\prime\prime}(0)=\frac{50}{(5+0)^{3}}=\frac{50}{125}=\frac{2}{5}).

Step4: Evaluate (f^{\prime\prime}(2))

Substitute (x = 2) into (f^{\prime\prime}(x)): (f^{\prime\prime}(2)=\frac{50}{(5 + 2)^{3}}=\frac{50}{343}).

Answer:

(f^{\prime\prime}(x)=\frac{50}{(5 + x)^{3}}); (f^{\prime\prime}(0)=\frac{2}{5}); (f^{\prime\prime}(2)=\frac{50}{343})