Find the shortest distance of the point (0, c) from the parabola `y=x^2` , where `0lt=clt=5` .

Explanation

If S is the distance of the point (x, y) on the parabola `y=x^2` from (0, c), then
`S=sqrt(x^2+(y-c)^2)=sqrt(x^2+(x-c)^2)` which is least `implies S^2` is least.
`frac{dS^2}{dx}=2x+2(x^2-c)times 2x=0`
`implies x=0` or `x^2 = frac{c-1}{2}`
When x=0, S=c. When `x^2=frac{c-1}{2}, cgefrac{1}{2}`
And `S=sqrt(c-frac{1}{2}+frac{1}{4})=sqrt(c-frac{1}{4})`
Hence the least distance is `c`
if `0leclefrac{1}{2}` and
`sqrt(c-frac{1}{4})`
if `frac{1}{2}lecle5`