Twenty metres of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum area (in sq. m) of the flower bed is:

The condition that $f(x) = ax^3 + bx^2 + cx + d$ has no extreme value is

If $f(x) = 1 + 2x^2 + 2^2 x^4 + \dots + 2^{10} x^{20}$,then $f(x)$ has:

$A$ rectangle $ABCD$ is inscribed in the region bounded by $y = \sin x$ and the $x-$axis for $x \in [0, \pi]$ (as shown in the figure). The area of the rectangle is maximum when $'\alpha'$ satisfies:

Find the local maximum and local minimum values for the function $h(x) = \sin x + \cos x$ in the interval $0 < x < \frac{\pi}{2}$.

Let $f(x)=3^{(x^{2}-2)^{3}+4}, x \in R$. Then which of the following statements are true?
$P: x=0$ is a point of local minima of $f$
$Q: x=\sqrt{2}$ is a point of inflection of $f$
$R: f^{\prime}$ is increasing for $x>\sqrt{2}$