If $$p(x)$$ be a polynomial of degree 3 satisfying $$p(-1)=10, p(1)=-6$$ and $$p(x)$$ has maximum at $$x=-1$$ and $$p'(x)$$ has minima at $$x=1$$. Find the distance between the local maximum and local minimum of the curve.

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).