Applications of Derivatives — Class 12 Mathematics PYQs

The absolute maximum value of function $f(x)=x^3-3x+2$ in $[0,2]$ is :

(A) $0$
(B) $2$
(C) $4$
(D) $5$

The function $f(x)=x^2-4x+6$ is increasing in the interval

(A) $(0,2)$
(B) $(-\infty ,2]$
(C) $[1,2]$
(D) $[2,\infty )$

A cylindrical tank of radius 10 cm is being filled with sugar at the rate of $100\pi$ cm${}^3$/s. The rate, at which the height of the sugar inside the tank is increasing, is :

(A) 0.1 cm/s
(B) 0.5 cm/s
(C) 1 cm/s
(D) 1.1 cm/s

Find the intervals in which function $f(x)=5x^{3/2}-3x^{5/2}$ is (i) increasing (ii) decreasing.

Find the values of '$a$' for which $f(x)=x^2-2ax+b$ is an increasing function for $x>0$.

Find the values of '$a$' for which $f(x)=\sqrt{3}\sin x-\cos x-2ax+b$ is decreasing on $\mathbb{R}$.

Find the values of '$a$' for which $f(x)=\sin x-ax+b$ is increasing on $\mathbb{R}$.

The side of an equilateral triangle is increasing at the rate of 3 cm/s. At what rate is its area increasing when the side of the triangle is 15 cm ?

The area of an expanding rectangle is increasing at the rate of 48 cm${}^2$/s. The length of the rectangle is always the square of its breadth. At what rate is the length of the rectangle increasing at an instant, when breadth $=4.5$ cm ?

A spherical medicine ball when dropped in water dissolves in such a way that the rate of decrease of volume at any instant is proportional to its surface area. Calculate the rate of decrease of its radius.