List of top Mathematics Questions on Application of derivatives asked in UP Board XII

Prove that the volume of the largest right circular cone that can be inscribed in a sphere of radius \(R\) is \(\dfrac{8}{27}\) of the volume of the sphere.

Prove that on \(R\), the function \(f(x)=e^{2x}\) is increasing.

If the radius of a circle is \(r = 6\) cm, then the rate of change of Area with respect to radius is:

Prove that the height of the cylinder of maximum volume inscribed in a right circular cone of semi-vertical angle \(\alpha\) and height \(h\) is one-third of the height of the cone, and the maximum volume of the cylinder is \(\dfrac{4}{27}\pi h^3\tan^2\alpha\).

The length \(x\) of a rectangle is decreasing at the rate of 3 cm/min and the breadth \(y\) is increasing at the rate of 2 cm/min. When \(x=5\) cm and \(y=3\) cm, find the rate of change of the area of the rectangle.

Prove that the height of the cylinder of maximum volume inscribed in a sphere of radius \(R\) is \(\dfrac{2R}{\sqrt3}\). Also find the maximum volume of the cylinder.

Prove that the radius of the largest (maximum surface area) right circular cylinder that can be inscribed in a cone is half of the radius of the cone.

A circular disc having radius 3 cm is warmed and due to expansion its radius is increasing at the rate 0.05 cm/s. Find the increasing rate of its area when its radius is 3.2 cm.

Prove that the given function \(f(x)=3x+17\) is increasing on \(\mathbb{R}\).

Prove that the height of a cylinder of maximum volume, inscribed in a sphere of radius \(R\), is \(\dfrac{2R}{\sqrt3}\).

Show that the function \(f(x)=x^{3}-6x^{2}+12x,\ x\in R\), is an increasing function on \(R\).

Prove that the height of the right circular cone of maximum volume inscribed in a sphere of radius \(r\) is \(\dfrac{4r}{3}\).

Find the interval in which the function \(f(x)=\dfrac{4\sin x-2x-x\cos x}{2+\cos x}\) is strictly increasing and strictly decreasing.

The interval in which the function \(f(x)=x^2-4x+6\) is increasing, is:

The radius of an air bubble is increasing at the rate of \(\frac{1}{2} \, \text{cm/s}\). At what rate is the volume of the bubble increasing while the radius is 1 cm?

Show that the function \( f(x) = 7x^2 - 3 \) is an increasing function when \( x>0 \).