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

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The function $f(x) = 2x^3 - 15x^2 + 36x + 5$ for $x \in [2,5]$ has

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The greatest possible value of '$a$' such that the function $f(x) = x^2 + a x + 1$ is always decreasing in the interval [1, 2] is:

The function $f(x) = \frac{x - 2}{x + 1}, x \neq -1$ is increasing when (Where $\mathbb{R}$ is a set of real numbers)

Consider the function $f(x) = \sin x$ in the interval $[\pi, 2\pi]$ then which of the following statements are correct?

(A) $x = \frac{3\pi}{2}$ is its stationary point.

(B) Its maximum value is 1

(C) Its minimum value is -1

(D) It attains its maximum value at $\pi$ and $2\pi$

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The radius of spherical balloon is decreasing at the rate of 0.1cm/sec, the rate at which its volume is decreasing, when its radius is 0.5cm is

Which of the following are NOT correct regarding the equation of tangent and normal to the curve $y = \frac{x-11}{(x-2)(x-3)}$ at the point, where it cuts the $x$-axis?

(A) The point of contact is (11, 0).

(B) The equation of tangent is $x - 72y - 11 = 0$

(C) The equation of normal is $72x + y - 11 = 0$

(D) The slope of the tangent at the given point of contact is $\frac{1}{88}$

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The function $f(x) = kx^3 + 6kx^2 + 18x + 17$ is increasing on $\mathbb{R}$(set of real numbers) if:

The total cost $c(x)$ associated with the production of $x$ units of an item is given by $c(x) = 0.001x^3 + 0.06x^2 + 20x + 500$. The marginal cost when 10 units are produced is:

The interval, on which the function $f(x) = x^2e^{-x}$ is increasing, is equal to

If the maximum value of the function $f(x) = \frac{\log_ex}{x}$, $x > 0$ occurs at $x = a$, then $a^2f''(a)$ is equal to

The rate of change of area of a circle with respect to its circumference when radius is 4cm, is

Match List-I with List-II

$\begin{array}{|l|l|} \hline \rule{0pt}{2.8ex}\text{List-I} & \text{List-II} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(A) The minimum value of } f(x) = (2x - 1)^2 + 3 & \text{(I) } 4 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(B) The maximum value of } f(x) = -|x + 1| + 4 & \text{(II) } 10 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(C) The minimum value of } f(x) = \sin(2x) + 6 & \text{(III) } 3 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(D) The maximum value of } f(x) = -(x - 1)^2 + 10 & \text{(IV) } 5 \\[1.2ex] \hline \end{array}$

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The function $f(x) = tanx - x$

The slope of the normal to the curve $y = 2x^2$ at $x = 1$ is:

The total cost $C(x)$ in Rupees associated with the production of $x$ units of an item is given by $C(x) = 0.007x^3 + 26x^2 + 15x + 400$. The marginal cost when 10 items are produced is:

The function $f(x) = \frac{x}{2} + \frac{2}{x}, x \neq 0$ is increasing on

(A) $(-\infty, -2)$

(B) $(-2, 2)$

(C) $(2, \infty)$

(D) $(-1, 1)$

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The absolute maximum value of the function $f(x) = 4x - \frac{1}{2}x^2$ in the interval $\left[-2, \frac{9}{2}\right]$ is

Match List-I with List-II

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The sum of two positive numbers is 60. If the sum of their squares in minimum, then the absolute value of the difference of their cubes is

The interval in which the function $g(x) = x^2 e^{-x}$ is increasing is:

The number of tangents to the curve $xy - 3y + 2 = 0$ having slope 2 is:

The function $f(x) = x^4 - 2x^2$ is increasing on

The function $f(x) = x + \frac{a^2}{x}$, $a > 0$, $x \neq 0$ has a local maxima at

Consider the function $f(x) = x^3 - 3x$. Then

Match List-I with List-II

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For the function $f(x) = sinx + cosx, x \in [0, \pi]$, which one of the following is correct?

In which of the following intervals, the function $f(x) = -x^2 - 2x + 15$ is decreasing?

In which of the following interval the function $f(x) = x^x, x > 0$ is strictly increasing?

If a function $f(x)=x^{2}+b x+1$ is increasing in the interval $[1,2]$, then the least value of $b$ is :

For the function $f(x) = 2x^3 - 9x^2 + 12x - 5$, $x \in [0, 3]$, match List-I with List-II :

The rate of change (in $\mathrm{cm}^{2} / \mathrm{s}$ ) of the total surface area of a hemisphere with respect to radius $r$ at $r=\sqrt[3]{1.331} \mathrm{~cm}$ is :