Question

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 $f^{\prime }(x)=0.$

• A. Statement- 1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement -1
• B. Statement -1 is True, Statement -2 is True ; Statement- 2 is NOT a correct explanation for Statement - 1
• C. Statement -1 is False, Statement -2 is True
• D. Statement - 1 is True, Statement- 2 is False

Answer 1

Answer: (D)

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 $z=y_0-\frac{1}{2}\le 0.$
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 $f^{\prime }\left(x\right)$ does not exist.