Gujarat Board-2018

The position of a particle is given by $S=f\left(t\right)=t^3-6t^2+9t,S$ is in metres, $t$ in seconds. Then find the distance travelled in first $5$ seconds.

Prove that $y=a{x}^3, x^2+3y^2=b^2$ are orthogonal.

A water tank is in the shape of an inverted cone. The radius of the base is $4\mathrm{m}$ and the height is $6\mathrm{m}$. The tank is being emptied for cleaning at the rate of $3{\mathrm{m}}^3/\min$. Find the rate at which the water level will be decreasing, when the water is $3\mathrm{m}$ deep.

Find the global maximum and minimum values of the function $f$ given by $f\left(x\right)=2x^3-15x^2+36x+1, x\in [1,5]$

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

A cylindrical can is to be made to hold $1 \mathrm{l}$ oil. Find its radius and height to minimize the cost.

Function $f\left(x\right)=\left|\sin x\right|, x\in \left(-\frac{\pi }{2},0\right)$ is:

Local maximum value of the $f\left(x\right)=x+\frac{1}{x}, (x\ne 0)$ is _____