The two parts of $100$ for which the sum of double of the first part and the square of the second part is minimum,are

$A$ triangular garden is fenced on two sides,and the third side is a straight river bank. The two fenced sides have an equal length of $x$. What is the maximum area that can be enclosed by the garden?

If $f(x) = 7e^{\sin^2 x} - e^{\cos^2 x} + 2$,then $\sqrt{7f_{\min} + f_{\max}}$ is equal to

$A$ cylindrical container is to be made from a certain solid material with the following constraints: It has a fixed inner volume of $V \ mm^3$,has a $2 \ mm$ thick solid wall,and is open at the top. The bottom of the container is a solid circular disc of thickness $2 \ mm$ and has a radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is $10 \ mm$,then the value of $\frac{V}{250 \pi}$ is

$P$ and $Q$ are two points on a circle with center $C$ and radius $\alpha$. The angle $\angle PCQ = 2\theta$. The radius $r$ of the circle inscribed in the triangle $CPQ$ is maximum when:

For the function $f(x) = x^{40} - x^{20}$,find the absolute minimum value in the interval $[0, 1]$.