Maxima and Minima

Find the maximum profit that a company can make, if the profit function is given by $p\left(x\right)=41+24x-18x^2$.

An enemy jet is flying along the curve $y=\left(x^2+2\right)$. A soldier is placed at the point $\left(3, 2\right)$. Find the nearest point between the soldier and the jet.

The least value of $f\left(x\right) =\left(e^x + e^{-x}\right)$ is$$

The maximum value of $f\left(x\right)=(x-2)(x-3{)}^2$ is$$

The minimum value of $f\left(x\right)=3x^4-8x^3+12x^2-48x+25$ on $\left[0,3\right]$ is$$

The minimum value of $f\left(x\right)=\left(x^2+\frac{250}{x}\right)$ is$$

If $x>0$ and $xy=1$, the minimum value of  $\left(x+y\right)$ is$$

$f\left(x\right)=\mathrm{cosec} x$ in $\left(-\mathrm{\pi },0\right)$ has a maxima at $$

The maximum Value of $\left(\frac{\log x}{x}\right)$ is $$

When $x$ is positive, the minimum value of $x^{\mathit{x}}$ is$$

$f\left(x\right)=\left|x\right|$ has 

Find the largest possible area of a right-angled triangle whose hypotenuse is $5 \mathrm{cm}$.

A wire of length $36 \mathrm{cm}$ is cut into two pieces. One of the pieces is turned in the form of a square and the other in the form of an equilateral triangle. Find the length of each piece so that the sum of the areas of the two be minimum.

An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of the material will be least when the depth of the tank is half of its width.

A square piece of tin of side $18 \mathrm{cm}$ is to be made into a box without the top, by cutting a square piece from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum? Also, find the maximum volume of the box.

A square tank of capacity $250$cubic meters has to be dug out. The cost of the land is Rs. $50$ per square metre. The cost of digging increases with the depth and for the whole tank, it is Rs.$\left(400\times h^2\right)$, where $h$ metres is the depth of the tank. What should be the dimensions of the tank so that the cost is minimum?

Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius $5\sqrt{3} \mathrm{cm}$ is $\left(500\mathrm{\pi }\right) {\mathrm{cm}}^3$

Two sides of a triangle have lengths $a$ and $b$ and the angle between them is $\theta$. What value of $\theta$  will maximize the area of the triangle?

A rectangle is inscribed in a semicircle of radius $\mathrm{r}$ with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle so that its area is maximum. Find also this area.

Show that the surface area of a closed cuboid with square base and given volume is minimum when it is a cube.