For the function $$f(x)=(\cos x)-x+1, x \in \mathbb{R}$$, between the following two statements

(S1) $$f(x)=0$$ for only one value of $$x$$ in $$[0, \pi]$$.

(S2) $$f(x)$$ is decreasing in $$\left[0, \frac{\pi}{2}\right]$$ and increasing in $$\left[\frac{\pi}{2}, \pi\right]$$.

The interval in which the function $$f(x)=x^x, x>0$$, is strictly increasing is

Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rectangle PQRS. Let a and b be the sides of the rectangle PQRS when its area is maximum. Then (a+b)$$^2$$ is equal to :

Let $$f(x)=x^5+2 x^3+3 x+1, x \in \mathbf{R}$$, and $$g(x)$$ be a function such that $$g(f(x))=x$$ for all $$x \in \mathbf{R}$$. Then $$\frac{g(7)}{g^{\prime}(7)}$$ is equal to :