Application of Derivative question bank (Maths)

Let $P$ be the point on the parabola $y = x^2$ such that the slope of the tangent to the parabola at the point $P$ is $4$. Let $Q$ be the point in the first quadrant lying on the circle $x^2 + y^2 = 2$ such that the slope of the tangent to the circle at the point $Q$ is $-1$. Let $R$ be the point in the first quadrant lying on the ellipse $x^2 + 4y^2 = 8$ such that the slope of the tangent to the ellipse at the point $R$ is $-\dfrac{1}{2}$.

Then the radius of the circle passing through the points $P, Q$ and $R$ is

Consider the function $f: (0, \infty) \rightarrow (-\infty, \infty)$ given by

$f(x) = \sqrt{x}\, \log_e(x) - x + 1$.

Then which one of the following statements is TRUE?