For a twice differentiable function $$f(x),g(x)$$ is defined as $$4\sqrt {65} g\left( x \right) = \left( {f'{{\left( x \right)}^2} + f''\left( x \right)} \right)\,\,f\left( x \right)$$ on $$\,\,\,\left[ {a,\,\,\,e} \right].$$ If for $$a < b < c < d < e,\,f\left( a \right) = 0,f\left( b \right) = 2,f\left( c \right) = - 1,f\left( d \right) = 2,f\left( e \right) = 0$$ then find the minimum number of zeros of $$g(x)$$.

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

If $$\left| {f\left( {{x_1}} \right) - f\left( {{x_2}} \right)} \right| < {\left( {{x_1} - {x_2}} \right)^2},$$ for all $${x_1},{x_2} \in R$$. Find the equation of tangent to the cuve $$y = f\left( x \right)$$ at the point $$(1, 2)$$.

Prove that for $$x \in \left[ {0,{\pi \over 2}} \right],$$ $$\sin x + 2x \ge {{3x\left( {x + 1} \right)} \over \pi }$$. Explain
the identity if any used in the proof.