Find the coordinates of the point in the graph of $f(x)=2x^2+3$ that is closer to the point $(5,-1)$
To start off, I found the first order derivative of the function so I could get the slope of the tangent line to $f(x)$
$f´(x)=4x$
Then, I needed to find the equation of the perpendicular line that goes through $(5,-1)$.
If $m_t*m_n=-1,$ therefore $m_n=-{1 \over4x}$
And then I proceeded to find the equation of the perpendicular line, getting as a result:
$y=-\frac 54+ {5 \over 4x} $
But as you can see, that's not a line. So I'm asking, what am I doing wrong? I'm thinking I have it wrong since I stated the slope of the tangent line is $4x$, I guessing I need a number.