List of top Mathematics Questions on Application of derivatives asked in JEE Advanced

Let \(\mathbb{R}\) denote the set of all real numbers. Consider the polynomial function \[ f : \mathbb{R} \to \mathbb{R} \] defined by \[ f(x) = \frac{d^{10}}{dx^{10}}\left((x^2 - 1)^{10}\right), \qquad \text{for all } x \in \mathbb{R}. \] Here, \[ \frac{d^{10}}{dx^{10}}\left((x^2 - 1)^{10}\right) \] is the \(10^{\text{th}}\) order derivative of the function \((x^2 - 1)^{10}\). Then which of the following statements is (are) TRUE?
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 \(-\frac12\). Then the radius of the circle passing through the points \(P\), \(Q\) and \(R\) is:
Consider the function \[ f:(0,\infty)\to(-\infty,\infty) \] given by \[ f(x)=\sqrt{x}\log_e(x)-x+1 \] Then which one of the following statements is TRUE?