CBSE Class 12 Mathematics — Continuity and Differentiability

(a) Check whether the function f(x) defined as $f(x) = \begin{cases} \dfrac{|x - 3|}{2(x - 3)}, & x < 3 \\ \dfrac{x-6}{6}, & x \ge 3 \end{cases}$ is continuous at $x = 3$ or not. OR (b) If $\sqrt{3} (x^2 + y^2) = 4xy$, then find $\dfrac{dy}{dx}$ at $\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$.

If $x = \cos t, y = \cos mt$, prove that $(1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} + m^2 y = 0$.