Determine the points on the parabola $y=x^2 - 25$ that are closest to $(0,3)$ 
Determine the points on the parabola $y=x^2 - 25$ that are closest to $(0,3)$

I would like to know how to go about solving this. I have some idea of solving it. I believe you have to use implicit differentiation and the distance formula but I don't know how to set it up. Hints would be appreciated.

So, $y=x^2-25$
So, we need to minimize $$f(x)=(x-0)^2+(x^2-25-3)^2=x^4-55x^2+28^2$$
$$\text{So,} f(x)=\left(x^2-\frac{55}2\right)^2+28^2-\left(\frac{55}2\right)^2\ge 28^2-\left(\frac{55}2\right)^2=\frac14$$ as $x$ is real
So, the distance will the minimum if $x^2-\frac{55}2=0\implies x=\pm\sqrt{\frac{55}2}, y=x^2-25=\frac{55}2-25=\frac52$
