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

If the minimum value of $a$ is $-\frac{k}{2}$ such that the function $f(x) = x^2 + ax + 5$ is increasing in [1, 2]. Then value of $k$ is

A spherical ice ball is melting at the rate of 100 $\pi$ cm³/min. The rate at which its radius is decreasing when its radius is 15 cm, is

For the function $f(x) = e^x + e^{-x}$

(A) $f'(x) = e^x - e^{-x}$

(B) The critical point is $x = 0$

(C) The minimum value is 1

(D) $x = 0$ is the point of local minimum.

Choose the correct answer from the options given below:

The equation of tangent line to $y = 2x^2 + 7$, which is parallel to the line $4x - y + 3 = 0$ is

If the cost function of a product is given by $C(x) = \frac{3}{4}x^2 - 5x + 21$, then the marginal cost when $x = 10$ is

The function $f: R \rightarrow R$ (where $R$ is set of real numbers) defined as

$f(x) = x^2 + 2x$ is

The point of local maxima of the function $f(x) = (x - 2)^5(x + 2)^2$ is

A car is moving along the curve $y = x^3 + 12$. The point(s) on the curve at which the rate of change of its y-coordinate at a certain time is 3 times the rate of change of its x-coordinate is/are

The rate of change of area of a circle with respect to its circumference when radius in 6 cm, is

Match List-I with List-II

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The largest open interval, in which the function $f(x) = \frac{x}{x^2 + 1}$ increases, is

The demand function (in Rs.) for a product is given by $P = 20 - 0.25x$, where P is the price per unit and x is the number of units sold, then the price of one unit, when the revenue is maximized, is:

Match List-I with List-II

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If $f(x) = 2x^3 - 15x^2 + 36x + 1$, $x \in [1, 5]$, then the absolute minimum value of $f(x)$ is:

The rate of change of volume of a sphere with respect to its surface area, when the radius is 6cm is:

The function $f(x) = x^2 - x + 1$ is

Let $f(x) = 4x^3 - 18x^2 + 27x - 5$, $x \in R$. Then which of the following statements are TRUE?

(A) $f''(x) = 24x - 36$

(B) f has local maxima at $x = \frac{3}{2}$ but no minima

(C) f has neither maxima nor minima

(D) f has both maxima and minima

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The function $f(x) = \frac{x}{3} + \frac{3}{x}$ is increasing in the interval:

The total cost function is given by $c(x) = \frac{1}{3}x^3 - 5x^2 + 30x - 15$ and selling price per unit is Rs.6. The profit is maximum if the value of x is:

The interval on which the function $f(x) = x^3 + 2x^2 - 1$ is decreasing, is

For the function $f(x) = e^{-2x}(2-x)^2$, the point of local maxima is:

For $x \in \mathbb{R}$, if $f(x) = -(x-1)^2 + 2$, then

(A) $f$ is an increasing function on $(-\infty, 1]$

(B) $f$ has no critical points

(C) $f$ has a maximum value at $x = 1$

(D) $f$ has a minimum value at $x = 1$

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A balloon which always remains spherical, has a variable diameter $\frac{3}{2}(5x+7)$. Then the rate of change of its volume with respect to x is

If $f(x) = \sin x - \cos x$, $x \in [0, 2\pi]$ then

(A) $f(x)$ is increasing in $(0, \frac{3\pi}{4})$

(B) $f(x)$ is decreasing in $(0, \frac{3\pi}{4})$

(C) $f(x)$ is decreasing in $(\frac{3\pi}{4}, \frac{7\pi}{4})$

(D) $f(x)$ is decreasing in $(\frac{7\pi}{4}, 2\pi)$

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The largest open interval in which the function $f(x) = 4x^3 - 5x^2 - 8x + 12$ increases, is:

A cylindrical drum of radius 7 cm and height 2 m is being kept in a vertical position filled with milk. If the milk is leaking at 14 cm³/sec from its lower base, then the rate of decrease in the level of milk is:

[Take $\pi = \frac{22}{7}$]

The function $f(x) = x^3 + 3x^2 + 4x + 4$, $x \in \mathbb{R}$ (set of real numbers) :

The maximum value of the function $f(x) = x^2(60 - x)$ in [20, 80] is:

The edge of a cube is increasing at a rate of 7 cm/s. The rate of change of area of the cube when its side is 3 cm is:

If it is given that at $x = 1$, the function $f(x) = x^4 - 62x^2 + 2ax + b$ attains its maximum value on the interval [0, 2], then the value of a is: