Application of Derivatives

On the interval $[0, 1]$, the function $f(x) = x^{25}(1 - x)^{75}$ attains its maximum value at the point $x = $....

If the line $ax + by + 5 = 0$ is a normal to the curve $xy = 1$ then .....

A spherical iron ball $10$ cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of $50\ \text{cm}^3/\text{min}$. When the thickness of ice is $5$ cm, the rate at which the thickness of ice decreases is...

The function $f(x) = x(x + 3)e^{-\left(\frac{1}{2}\right)x}$ satisfies all the conditions of Rolle's theorem in $[-3, 0]$, then $c =$

The line $x + y = 0$ touches the curve $y^2 = ax^3 + b$ at $(1, -1)$ then values of $a$ and $b$ respectively are ...........

An aeroplane at an altitude of 1 km is flying horizontally at 600 km / hr, passes directly over an observer. Then the rate at which it is approaching the observer when it is 1250 meters away from him is........

The value of $x$ so that the volume of the parallelopiped formed by the vectors $\hat{i} + x\hat{j} + \hat{k}$, $\hat{j} + x\hat{k}$ and $x\hat{i} + \hat{k}$ is minimum, is

The value of $c$ satisfied by the Rolle's theorem for the function $f(x) = x^2(1 - x)^2$, $x \in [0, 1]$ is...

A particle is fired straight up from the ground. Its height in feet after $t$ second is given by $s(t) = 128t - 16t^2$. The velocity of the particle when it hits the ground is...

A tank with a rectangular base and rectangular sides, open at the top is made. Depth of the tank is $4$ m and its volume is $36$ cubic meters. For making a tank cost of base material used is Rs. $100$ per sq. meter and that of sides is Rs. ...

If the rate of increase of surface area of a spherical balloon is $5\,\text{cm}^2/\text{sec}$ and rate of increase of volume of a spherical balloon is $10\,\text{cm}^3/\text{sec}$, then the radius of the balloon at that time is...

Rolle's theorem holds for monic quadratic polynomial $f(x)$ on the interval $[\alpha, \alpha + 3]$ where $f(\alpha) = 0$. Similarly, $g(x) = f(x) + 2$ also follows Rolle's theorem in the interval $[\beta, 3]$ where $g(3) = 0$, such that the...

Let PA and PB be the tangent segments drawn from point P$(6, 8)$ to the circle with the centre at origin O. The radius of circle for which the area of quadrilateral PAOB is maximum, is...

If the side of an equilateral triangle increases at the rate of $\sqrt{3}\ \text{cm/sec}$, then the rate of change of increase of its area when the side is $12\ \text{cm}$ is ____

If the tangent to the curve $xy + ax + by = 0$ at $(1,1)$ makes an angle of $\tan^{-1}2$ with positive direction of the $x$-axis, then the value of $\dfrac{ab}{a+b}$ is...

If the line $x + By + C = 0$ is the normal to the curve given by $x = a\sin^3 t$, $y = b\cos^3 t$, (where $a, b \neq 0$) at a point $t = \dfrac{\pi}{2}$, then $B - C = $

If a particle moves such that the displacement (s) is proportional to the square of the velocity (v), then its acceleration (a) is

If the function $f(x) = ax^2 + bx + \sin x$ satisfies all the conditions of Rolle's theorem on $[0, \pi]$ and the slope of the tangent to the curve $y = f(x)$ at $x = \dfrac{\pi}{4}$ is zero, then $a - b = $

Let $g(x) = f(x) + f(1-x)$ and $f''(x) < 0, 0 \leq x \leq 1$, then $\ldots$

The number 28 is divided into two positive parts such that the sum of the cube of one part and the square of the other part is minimum, then the absolute difference between the two parts is

If $f(x) = \log(1 + x) - \dfrac{x}{1+x}$, then the values of $x$ for which $f(x)$ is monotonically increasing and monotonically decreasing are respectively.....

The function $f(x) = \int \dfrac{x+3}{x^2-9x+20}\,dx$, then $f(x)$ is

The derivative of the function $f(x) = \cos^4 x + \sin^4 x,\ 0 \leq x \leq 2\pi$ is positive for

If the line $y = 4x - 5$ is tangent to the curve $y^2 = ax^3 + b$ at the point $(2,3)$, then the value of $7a - 2b$ is...

The equation of the tangent to the curve $y = \sqrt{9 - 3x^2}$ at the point where the ordinate and abscissa equal is...

The coordinates of the points on the curve $4y = x^2$ that are nearest to the point $(0,5)$ are ...

The minimum value of $\dfrac{\log x}{x}$ in the interval $(2, \infty)$ is

The function $f(x) = \tan^{-1}(\sin x + \cos x)$ is an increasing function in the interval.....

A ball is thrown in the air. Its height at any time $t$ is given by $h = 3 + 14t - 5t^2$, then the maximum height it can reach

The equation of tangent to the curves $x = 1 - 3t^2$ and $y = t - 3t^3$ at the point $(-2, 2)$ is...

A spherical snow ball is melting so that its volume is decreasing at the rate of 8 c.c./sec then the rate of change of radius when the radius is 2 cm, is :

A cylindrical tank without a top lid is being manufactured to hold a fixed volume of $125\pi$ cubic cm. The minimum surface area required to construct this tank is ............square cm

The co-ordinates of the point on the curve $y = x\log x$ at which the normal is parallel to the line $2x - 2y = 3$ are...

If the function $f(x) = ax^3 - bx^2 - 8x - 4$ satisfies Roll's theorem in $[1,3]$, if $f'(2) = 0$ then $a - b$ is equal to...

The point on the curve $9y^2 = x^3$ where the normal to the curve makes equal intercepts with the co-ordinate axes is

The maximum value of $\left(\dfrac{1}{x}\right)^x$, $x > 0$ is

The surface area of a spherical ball is increasing at the rate of $4\pi\ \text{cm}^2$/second. The rate at which the radius is increasing when the surface area is $16\pi\ \text{cm}^2$ is

The approximate value of $(0.007)^{\frac{1}{3}}$ is

If a spherical balloon has a variable diameter $3x + \dfrac{9}{2}$ units, then the rate of change of its volume with respect to $x$ is

A wire 40 metre in length is to be cut into two pieces. One piece is formed into a square and the other piece into a circle. The lengths of the two pieces so that the combined area of the square and the circle is minimum, are respectively

The equation of the normal to the curve $xy + 7 = 0$ is $Ax + By + C = 0$, then

A point on the parabola $y^2 = \dfrac{36}{5}x$ at which the ordinate increases at thrice the rate of the abscissa is .....

Let $f(x)$ be the differentiable function for all $x$ such that $f'(x) \leq 5$ and $f(1) = 4$. The maximum value of $f(5)$ is...

If the tangent to the curve $2y^3 = x^3 + ax^2$ at the point $(a, a)$ cuts off intercepts $\alpha$ and $\beta$ on the coordinate axes such that $\alpha^2 + \beta^2 = 61$, then the value of $a$ is

If Rolle's theorem is applicable for the function $f(x) = \log\left(\dfrac{x^2+a}{x}\right)$ on $[3, 4]$ with $c \in (3, 4)$ such that $f'(c) = 0$, then the value of $f''(c)$ is

The equation of the tangent to the curve $y = 3x^3 - 3x^2 + x$ at $x = 1$ is

The approximate value of $\sqrt[3]{63}$ is

A wire of length $64$ m is to be bent to form a rectangle such that its area is maximum, then its area is.....

The difference between the local extreme values of the function $f(x) = 2x^3 - 15x^2 + 36x + 40$ is ......

The value of $k$ such that the function $f(x) = \sin x - \cos x - kx + b$ is strictly decreasing for all real $x$ is