We got two functions:
- $f(x)=ax^2+b$
- $g(x)=x^{-1}=1/x$
I know that they are touching each other in $x=1$. Now I can find out the values for $a$ and $b$ in $f(x)$.
- Set the derivative of both functions equal $f'(1)=g'(1)$ to get $a$ $\begin{align}&f'(x)=2ax; g'(x)=x^{-2}\\ \implies& 2a(1)=(1)^{-2}\\ \implies& a = -\frac12\end{align}$
- Set the base functions equal $f(1)=g(1)$ to get $b$
$\begin{align}&f(x)=-\frac12x^2+b\\ \implies & -\frac12(1)+b=(1)^{-1}\\ \implies & b = \frac32\end{align}$ 3. Control the result (this is were my issue is) $\begin{align}&f(x)=g(x)\\ \implies&-\frac12x^2+\frac32=x^{-1} &|& \text{ subract } x^{-1}\\ \Longleftrightarrow& -\frac12x^2-x^{-1}+\frac32=0 &|&\text{ multiply by } -2\\ \Longleftrightarrow& x^2-2x^{-1}-3=0\end{align}$
I now would like to get $x=1$ to control mathematically if my above result is valid.
But I have no idea how I could solve a function with two powers