If $y=a \log x+b x^2+x$ has its extremum values at $x=-1$ and $x=2$,then

Let $f(x) = x^3 + px + 1$ and consider the following three statements:
$(i)$ For $p \geqslant 0$,$f(x) = 0$ has one negative root and $f(x)$ is monotonic.
$(ii)$ For $-1 < p < 0$,$f(x) = 0$ has one negative root and $f(x)$ is non-monotonic.
$(iii)$ For $p < -3/\sqrt[3]{4}$,$f(x) = 0$ has three real and distinct roots.
Which of the following is correct?

The maximum area of a right-angled triangle with hypotenuse $h$ is

The area bounded by the tangents of the curve given by $y = \sin \theta \cos^2 \theta$ and $x = \sin^2 \theta \cos \theta$,which are parallel to the coordinate axes (excluding the axes themselves),is:

For the function $f(x)=x^3-6x^2+12x-3$,the point $x=2$ is

The sum of the perimeter of a circle and a square is $k$,where $k$ is a constant. Prove that the sum of their areas is least when the side of the square is double the radius of the circle.