If $$f(x)$$ is a differentiable function and $$g(x)$$ is a double differentiable function such that $$|f(x)| \leq 1$$ and $$f'(x)=g(x)$$, where,$$f^{2}(0)+g^{2}(0)=9$$ then prove that there exists some $$c \in(-3,3)$$ such that $$g(c) \circ g^{n}(c) < 0$$.

If $$\left[\begin{array}{lll}4 a^{2} & 4 a & 1 \\ 4 b^{2} & 4 b & 1 \\ 4 c^{2} & 4 c & 1\end{array}\right]\left[\begin{array}{c}f(-1) \\ f(1) \\ f(2)\end{array}\right]=\left[\begin{array}{c}3 a^{2}+3 a \\ 3 b^{2}+3 b \\ 3 c^{2}+3 c\end{array}\right], \quad f(x)$$

is a quadratic function and its maximum value occurs at a point $$\mathrm{V}$$. If A is a point of intersection of $$y=f(x)$$ with $$x$$-axis and point B is such that chord AB subtends a right angle at point $$\mathrm{V}$$. Find the area enclosed by $$f(x)$$ and chord AB.

A whistling train approaches a junction. An observer standing at the junction observes the frequency to be 2.2 kHz and 1.8 kHz of the approaching and the receding train. Find the speed of the train (speed of sound = 300 m/s).

A conducting liquid bubble of radius $$a$$ and thickness $$t(t < < a)$$ is charged to potential V. If the bubble collapses to a droplet, find the potential on the droplet.