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

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

The point on the curve $\frac{x^2}{4} + \frac{y^2}{9} = 1$ at which the tangent to the curve is parallel to the x-axis is

The demand for a certain product is represented by the function $p = 300 + 25x - x^2$ (in rupees), where x is the number of units demanded and p is the price per unit, then the marginal revenue when 15 units are sold, is

If $f(x) = x^2 - 4x + 13, x \in \mathbb{R}$, then which of the following are correct?

(A) $x = 2$ is a stationary point of $f(x)$.

(B) $f(x)$ is increasing function on $(2, \infty)$

(C) $f(x)$ have maxima at $x = 2$

(D) $f(2) = 9$

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Function $f(x) = x^3 - 3x + 3$ is

(A) Increasing in the interval $(-1, 1)$

(B) Increasing in the interval $(1, \infty)$

(C) Decreasing in the interval $(-1, 1)$

(D) Increasing in the interval $(-\infty, -1) \cup (1, \infty)$

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The function $f(x) = \sin 3x$, $x \in \left[0, \frac{\pi}{2}\right]$

(A) is increasing on $\left[0, \frac{\pi}{6}\right]$

(B) is decreasing on $\left[\frac{\pi}{6}, \frac{\pi}{2}\right]$

(C) is increasing on $\left[0, \frac{\pi}{2}\right]$

(D) is decreasing on $\left[0, \frac{\pi}{2}\right]$

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The semi vertical angle of a right circular cone of maximum volume of a given slant height is

The minimum value of the function $f(x) = x^3 + (10-x)^3$ occurs at:

If a revenue function is given by $R(x) = 2027x - 1013x^2 - 675x^3$, then the marginal revenue function (MR) is:

For the function $f(x) = x^{1/x}$, $x > 0$, which of the following are correct?

(A) $x = 0$ is the only point where extremum may occur.

(B) The given function is maximum at $x = e$.

(C) The function has no extreme value for $x > 0$.

(D) The maximum value of the function $f(x)$ is $e^{1/e}$.

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Match List-I with List-II

Consider the function f(x) = 2x³ - 21x² + 36x + 80, x∈[0, 6]. Then

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If the interval in which f(x) = $\frac{x}{4}$ + $\frac{4}{x}$, x ≠ 0 is strictly increasing is (-∞, a) ∪ (b, ∞), then

For x ∈ ℝ - {0}, the function f(x) = $\frac{3}{x}$ + 7 is decreasing when

The maximum value of f(x) = $\left(\frac{1}{x}\right)^x$ is

If the interval in which the function f(x) = $\frac{x}{x^2+1}$ is strictly increasing is (-a, a), then a is equal to

The point on the curve y = (x - 2)² at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4) is:

The real valued function $f(x) = 12x^\frac{4}{3} - 6x^\frac{1}{3}, x \in [-8, 8]$ has absolute maximum value equal to

If the interval in which the function $f(x) = 4x^3 - 6x^2 - 72x + 30$ is strictly decreasing, is (a,b) then a+b is equal to

The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. The rate at which the area increases when the side is 10 cm, is

Match List-I with List-II

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

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The interval on which the function $f(x) = x^4 - \frac{x^3}{3}$ is strictly decreasing, is:

The equation of the tangent to the curve $y = \frac{(x - 3)}{(x - 1)(x - 2)}$ at the point, where it cuts x-axis is:

In which of the following interval, the function $f(x) = \frac{x}{\log x}$ is decreasing?

If $f(x) = a \log_e|x| + bx^2 + x$ has critical points at $x = -2$ and $x = 1$, then

The function $f(x) = x^2 - 4x + 6$ is

(A) Strictly decreasing on $(-\infty, 2) \cup (2, \infty)$

(B) Strictly increasing on $(2, \infty)$

(C) Strictly increasing on $(-\infty, \infty)$

(D) Strictly decreasing on $(-\infty, 2)$

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Which of the following functions are increasing on $x \in \left(0, \frac{\pi}{2}\right)$?

(A) $f(x) = \sin x$

(B) $f(x) = \cos x$

(C) $f(x) = \tan x$

(D) $f(x) = \cos 3x$

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The maximum value of $f(x) = \frac{1}{4x^2 + 2x + 1}$ is

If the area of an equilateral triangle is increasing at the rate of $4\sqrt{3}$ cm²/sec, then the rate of increase of its perimeter when the side is 4cm, is

Which of the following are correct?

(A) The function $f(x) = 3x+12$ is increasing on R.

(B) The function $f(x) = e^{2x}$ is decreasing on R.

(C) The function $f(x) = x^2-x-1$ is neither increasing nor decreasing on (-1, 1).

(D) The function $f(x) = x^3-3x^2+4x$ is increasing on R.

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If the function $f(x) = 2x^3 + 9x^2 + 12x-1$ is given,then $f(x)$ have