Application of Derivatives

If $f(x)=a \log |x|+b x^2+x$ has extreme values at $x=-1$ and $x=2$, then the ordered pair $(a, b)=$

Maximum area of the rectangle inscribed in a circle of radius $10 \mathrm{cms}$ is

If the height of a cone of greatest volume that can be inscribed in a sphere of radius $\mathrm{R}$ is $\mathrm{kR}$, then ratio of the volume of the cone to the volume of the sphere is

Given that the solid obtained by rotating a rectangle about one of its side is a cylinder. If the perimeter of a rectangle is $48 \mathrm{~cm}$ and the volume of the cylinder formed by rotating it is maximum, then the…

An angle between the curves $x^2=3 y$ and $x^2+y^2=4$ is

If a cylindrical vessel of given volume $V$ with no lid on the top is to be made from a sheet of metal, then the radius $(r)$ and height $(h)$ of the vessel so that the metal sheet used is minimum, is

Let $f(x)=\operatorname{Max}\{\cos x, \sin x, 0\}$. If the number of points at which $f(x)$ is not differentiable in $(0,2024 \pi)$ is $1012 k$, then $\mathrm{k}=$

If a running track of 500 ft . is to be laid out enclosing a playground, the shape of which is a rectangle with a semicircle at each end, then the length of the rectangular portion such that the area of the rectangular…

The angle between the curve $2 y=e^{-x / 2}$ and the $y$-axis is $\tan ^{-1}(\mathrm{k})$ then $\mathrm{k}=$

In the interval $\left(\frac{1}{\mathrm{e}}, \mathrm{e}\right)$, a decreasing function among the following functions is

The minimum distance of a point on the curve $y=x^2-4$ from the origin is

If $\beta$ is an angle between the normals drawn to the curve $x^2+3 y^2=9$ at the points $(3 \cos \theta, \sqrt{3} \sin \theta)$ and $(-3 \sin \theta, \sqrt{3} \cos \theta), \theta \in\left(0, \frac{\pi}{2}\right)$,…