CUET Mathematics: CalculusApplication of Derivatives. Free, no login required.

Let $f(x)$ be a function defined on an interval $[a,b]$ such that $f(x)$ is differentiable on $(a,b)$. Then

Match the LIST-I with LIST-II

Choose the correct answer from the options given below:

The function $f(x) = 2x^3 - 3x^2 - 36x + 7$ is :

A. Increasing in $(-\infty,-2)$

B. Decreasing in $(-2,3)$

C. Neither increasing nor decreasing in $(-2,3)$

D. Increasing in $(3,\infty)$

Choose the correct answer from the options given below:

If the function $f(x) = x^4 - 62x^2 + ax + 9$ has a critical point at $x = 1$, then $a$ is equal to:

The ordinate of the point on the curve $y^2 = 8x$ for which abscissa and ordinate change at the same rate is:

The rate of change in area of a triangle having sides 10cm and 12cm when the variable angle between them at an angle $60°$, is:

The function $f(x) = 24x^4 - 32x^3 - 96x^2 + 7$ on the interval $\left[-\dfrac{3}{2}, 4\right]$ is :

A. increasing in $(-1,0)$

B. decreasing in $(0,4)$

C. increasing in $(2,4)$

D. decreasing in $\left(-\dfrac{3}{2},-1\right)$

Choose the correct answer from the options given below:

Let $f(x)$ be a function defined by $f(x) = 2x^3 - 24x + 107$. If $\alpha$ and $\beta$ are the maximum and minimum values respectively in the interval $[1,3]$, then $\alpha-\beta$ is equal to:

The rate of change of $\sqrt{x^2-2}$ with respect to $\dfrac{x}{x+1}$ at $x=2$ is :

If $f(x) = (x+3)e^{-x}$, then which of the following statements are True?

(A) Function $f(x)$ is increasing on $(-2, \infty)$.

(B) Function $f(x)$ is decreasing on $(-2, \infty)$.

(C) Function $f(x)$ is increasing on $(-\infty, -2)$.

(D) Function $f(x)$ is decreasing on $(2, 3)$.

The local minimum value of $f(x) = x^3 + \frac{243}{x}$ is

The side of an equilateral triangle is increasing at the rate of 2 cm/sec. The rate at which the area of the triangle is increasing, when the side of the triangle is 5 cm, is

For the function $f(x) = x^5 - 5x^4 + 5x^3 - 1$, which of the following are true?

(A) $f(x)$ has local minimum value equal to -28.

(B) $f(x)$ has point of inflection at $x=1$.

(C) $f(x)$ has point of local maxima at $x=1$.

(D) $f(x)$ has the local maximum value equal to 2.

Choose the correct answer from the options given below:

A closed right circular cylinder has a volume of 2156 cm³. The radius of its base so that its total surface area may be minimum, is equal to:

For which of the following real functions, $x=0$ is a point of local maxima?

(A) $f(x)=4x^4-3x^3-2x^2+1$

(B) $f(x)=3x^5+\sqrt{2}x^4-7x^2+4$

(C) $f(x)=5x^4-3x^2+e$

(D) $f(x)=-3x^3+2x^2-9$

Choose the correct answer from the options given below:

Let $f(x) = (x-a)^2+(x-b)^2+(x-c)^2$ then $f(x)$ has a minimum value at $x =$

The function $f(x) = -x^3 + 12x^2 - 36x + 21$ is

A. Increasing on $(-\infty, 2)$

B. Decreasing on $(-\infty, 2)$

C. Increasing on $(2, 6)$

D. Decreasing on $(6, \infty)$

Choose the correct answer from the options given below:

A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimeters of gas per second. The rate at which the radius of the balloon increases when the radius is 15 cm, is:

In a sphere of radius $r$, a right circular cone of height $h$ having a maximum curved surface area is inscribed. The expression for the curved surface area of the cone is

The function $f(x)=\cot^{-1}x+x$ increases in the interval:

The function $f(x)=17-9x+6x^2-x^3$ is strictly increasing in the interval, is:

If the total cost function is given by $C(x)=\frac{x^3}{3}-2x^2+9x-70$ and the selling price per unit is ₹6, then for what value of $x$ will the profit be maximum?

For the function $f(x) = (x-1)^3(x-2)^2$

Match the LIST-I with LIST-II

Choose the correct answer from the options given below:

A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lower most. Its semi vertical angle is $\tan^{-1}\left(\frac{2}{3}\right)$. Water is poured into it at a constant rate 6 cubic metre per hour. The rate at which the level of the water is rising at the instant when the depth of water in the tank is 5 meter is

A function $f(x)=\sin x(1+\cos x), x \in [0,\pi]$ has a local maxima at $x=$

Function $f(x)=\sin3x, x \in \left[0,\frac{\pi}{2}\right]$ is decreasing in the interval

The interval in which the function $f(x)=x^3-12x^2+36x+20$ is increasing, is

For the function $f(x)=2x^3-9x^2+12x-5, x \in [0,3]$, which of the following statements are correct ?

A. The point of absolute maxima is x=3.

B. The absolute minimum value is -1.

C. The critical points are x=1 and x=2.

D. The absolute maximum value is 3.

Choose the correct answer from the options given below:

The function $f(x)=81-3x^3$ has

Rate of change of area of a circle with respect to radius

(A) When r = 2 cm is $4\pi$ cm²/cm.

(B) When r = 6 cm is $12\pi$ cm²/cm.

(C) When r = 12 cm is $6\pi$ cm²/cm.

(D) When r = 16 cm is $8\pi$ cm²/cm.

Choose the correct answer from the options given below:

An open box is made from a cardboard measuring 12cm x 12cm by cutting off equal squares from the corner and turning up the side. Then maximum possible volume of the box would be: