Question

Find the shortest distance from the origin to the parabola y=4-x^2

Answer 1

let (x,y) be the point that yields the minimum distance from the origin

Answer 2

\[D^2=(x-x_0)^2+(y-y_0)^2\]
\[(x_0,y_0) \implies (0,0)\]
\[D^2=(x-0)^2+(y-0)^2=x^2+y^2\]

Answer 3

since \[y=4-x^2\]
then
\[y^2=(4-x^2)^2=16-8x^2+x^4\]
so\[D^2(x)=x^4-8x^2+16+x^2=x^4-7x^2+16\]
minimise \[D^2(x)\]
by
finding the derivative