$(6$ points$)$ Consider the following function

Find the Taylor polynomial $T_2(x)of$ degree $2$ near $x=0off(x).$

$f(a)=\frac{3}{2}(1+x{)}^{\frac{1}{2}}=x_n,x^2=f(0)+\frac{f(0{)}^{\text{'}}}{1!}x-a+\frac{f^{\text{'}\text{'}}(0)}{2!}(x-a{)}^2+\frac{f^{\text{'}\text{'}\text{'}}(c)}{3!}$

$f^{\text{'}}(0)=\frac{1}{e^{2a}}=(\frac{3}{2})(\frac{1}{2})(1+x)=Z(1+x)!T_2=f(a)+\frac{f(a{)}^{\text{'}}}{1!}x+\frac{f(a{)}^{\text{'}\text{'}}}{2!}{x}^2$

$f^{\text{'}\text{'}}(x)=\frac{1}{2e}=4(1+x^2)!$

$T_2=3e^{x^2}+\frac{1}{e^{2x}x}+\frac{1}{2!}{x}^2$

$T_2=\frac{3}{2}(1+x{)}^{\frac{1}{2}}+\frac{2(1+x)!}{1!}x+\frac{4(1+x^2)!}{2!}{x}^2$