Difference between the maximum and minimum values of $f(x)=-\sin ^3 x+3 \sin ^2 x+5$ in $x \in\left[0, \frac{\pi}{2}\right]$ is

If tangent to the curve $f(x)=x^3-\alpha x^2-x+\beta$ at point $(1,3)$ on the curve, cut equals non zero intercepts on co-ordinate axes, then

The maximum slope of the curve $$y=\frac{1}{2} x^4-5 x^3+18 x^2-19 x+7$$ occurs at the point

$$f$$ and $$g$$ are differentiable function in $$(0,1)$$ satisfying $$f(0)=2=g(\mathrm{l}), g(0)=0$$ and $$f(l)=6$$, then for some $$c \in] 0,1[$$