1. Tardigrade
  2. Question
  3. Mathematics
  4. Statement-1: The minimum distance of the fixed point (0, y0) where 0 le y0 < (1/2), from the curve y=x2 is y0. Statement-2: Maxima and minima of a function is always a root of the equation f ' (x) = 0.

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Q. Statement-1 : The minimum distance of the fixed point $\left(0, y_{0}\right)$ where $0 \le y_{0} < \frac{1}{2}$, from the curve $y=x^{2}$ is $y_{0}.$
Statement-2 : Maxima and minima of a function is always a root of the equation $f ' (x) = 0.$

Application of Derivatives

Solution:

Let the point on the parabola be $(t, t_2).$
Let d be the distance between $(t, t^2)$ and $(0, y_0)$, then
$d^2 = t^2 + (t^2 - y_0)^2 = t^4 + (1 - 2y_0) t^2 + y_0 ^2$
$= z^2 + (1 - 2y_0) z + y_0 ^2, z \ge 0$
its vertex is at $z=y_{0}-\frac{1}{2} \le0.$
the minimum value of $d^{2}$ is at $z = 0$
i.e. t$^{2}=0$
$\therefore d=y_{0}$
$\therefore $ Statement 1 is true. Statement 2 is false because extremum can occur at a point where $f '\left(x\right)$ does not exist.