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

If the sum of the lengths of hypotenuse and a side of a right angled triangle is given, then the area of the triangle is maximum when angle between them is:

The absolute maximum value of the function $f(x) = 2x^3 - 15x^2 + 36x + 7$ on the interval [1,5] is

The interval in which the function $f(x) = \log_e(2+x) - \frac{2x}{2+x}$ is increasing is

For the function $f(x)=2x^3-21x^2+36x-100$;

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If $\alpha$ and $\beta$ are absolute maximum and absolute minimum values of a function $f(x)=\dfrac{x}{2}+\dfrac{2}{x}$ on the interval $[1,2]$, then the value of $\alpha+\beta$ is:

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The function $f(x)=\sin^4(x)+\cos^4(x)$, on interval $\left(0,\dfrac{\pi}{2}\right)$, has

The length of a rectangle is decreasing at 4 cm/minute and the width is increasing at 3 cm/minute. The rate of change of area when length is 10cm and width is 6cm, is:

For the function $f(x)=x^3-2x^2-4x-5$, which of the following statement is not correct ?

The total revenue received from the sale of $x$ units of a product is given by $R(x)=542+\dfrac{x^2+3}{4}$. What will be the marginal revenue when $x=24$ ?

The interval in which the function $f(x)=x^4-\dfrac{x^3}{3}$ is increasing, is

If $y = a\log x + bx^2 + x$ has extreme values at $x = 1$ and $x = 2$, then

The function $f(x) = \frac{4x^2+1}{x}$ is decreasing in

(A). $\left(\frac{-1}{2}, \frac{1}{2}\right)$

(B). $\left(-\infty, \frac{-1}{2}\right)$

(C). $\left(0, \frac{1}{2}\right)$

(D). $\left(\frac{-1}{2}, 0\right)$

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A solid sphere of gold of radius 5 cm, is being melted in a furnace such that the radius is decreasing uniformly. When the radius is 1cm, the rate at which surface area (A) is decreasing w.r.t. time ($t$) is equal to

($\frac{dv}{dt}$ = rate at which volume is decreasing w.r.t. $t$)

If the function $f(x) = x^2-2kx+7$ is increasing on [1, 5] then $k$ belongs to

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

The function $f(x) = 2x^3-15x^2+36x+4$ has local maxima at $x=$

If $f(x) = x^4 - \dfrac{2x^3}{3}$, then $f(x)$ is

A. increasing in $\left(\dfrac{1}{2}, \infty\right)$

B. decreasing in $\left(\dfrac{1}{2}, \infty\right)$

C. increasing in $(-\infty, 0)$

D. decreasing in $(-\infty, 0) \cup \left(0, \dfrac{1}{2}\right)$

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Maximum slope of the curve $y = -x^3 + 3x^2 + 9x - 30$ is

If an open box with a square base is to be made out of a given card board of area $K^2$ square units, then the maximum volume of the box is

For the function $f(x) = \dfrac{3}{2}x^4 - 4x^3 - 45x^2 + 49$. Which of the following statements is/are TRUE ?

A. $f(x)$ is increasing in $(-\infty, -3)$

B. $f(x)$ is increasing in $(-3, 0) \cup (5, \infty)$

C. $f(x)$ is decreasing in $(-\infty, -3) \cup (0, 5)$

D. $f(x)$ is decreasing in $(-\infty, -3) \cup (5, \infty)$

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Water is running into a conical vessel, 18cm deep and 6cm in radius, at the rate of 0.1 $cm^3$/sec. If the water is 4cm deep, then the water level rising at the rate

The demand function for a monopolist is given by $x = 100 - 4p$, then the marginal revenue, when $x=6$ is (where $x$ is the number of units and $p$ is the price per unit)

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Match the LIST-I with LIST-II

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For the function $f(x) = \frac{1}{x^2+2x+2}$

which of the following statements are TRUE?

A. $f(x)$ is increasing on $(-\infty, -1)$

B. $f(x)$ is decreasing on $(-\infty, -1)$

C. $f(x)$ is decreasing on $(-1, \infty)$

D. maximum value of $f(x)$ is 1

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The minimum value of $f(x) = e^x + e^{-x}$ is

In a sphere the rate of change of volume with respect to(w.r.t) time is

The maximum value of the function $f(x)=\sin x(1+\cos x)$ in the internal $[0,\pi]$ is

For the function $f(x)=x^4-2x^2+5, x\in R$, then which of the following statements are True?

A. $f(x)$ is increasing on $(-\infty,-1)$

B. $f(x)$ is decreasing on $(-\infty,-1)$

C. $f(x)$ is increasing on $(-1,0)\cup(1,\infty)$

D. $f(x)$ is decreasing on $(1,\infty)$

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