Application of Derivatives

The abscissa of the points, where the tangent to the curve \(y=x^3-3 x^2-9 x+5\) is parallel to \(X\) axis are

The equation of the tangent to the curve \(y = 4x{e^x}\) at \(\left( { - 1,{{ - 4} \over e}} \right)\) is

A particle is moving on a straight line. The distance \(\mathrm{S}\) travelled in time \(\mathrm{t}\) is given by \(\mathrm{S=a t^2+b t+6}\). If the particle comes to rest after 4 seconds at a distance of \(16 \mathrm{~m}\). from the starti...

A wire of length 20 units is divided into two parts such that the product of one part and cube of the other part is maximum, then product of these parts is

Water is being poured at the rate of 36 m\(^3\)/min. into a cylindrical vessel, whose circular base is of radius 3 m. Then the wate level in the cylinder is rising at the rate of

The equation of tangent to the circle \(x^2+y^2=64\) at the point \(\mathrm{P\left(\frac{2\pi}{3}\right)}\) is

If \(f(x)=2x^3-15x^2-144x-7\), then \(f(x)\) is strictly decreasing in

The surface area of a spherical balloon is increasing at the rate \(2 \mathrm{~cm}^2 / \mathrm{sec}\). Then rate of increase in the volume of the balloon is , when the radius of the balloon is \(6 \mathrm{~cm}\).

The area of the square increases at the rate of $0.5 \mathrm{~cm}^2 / \mathrm{sec}$. The rate at which its perimeter is increasing when the side of the square is 10 cm long, is

The equation of the normal to the curve $2 x^2+y^2=12$ at the point $(2,2)$ is

The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is

The equation of tangent at \(P(-4,-4)\) on the curve \(x^2=-4 y\) is

If \(f(x)=\log (\sin x), x \in\left[\frac{\pi}{6}, \frac{5 \pi}{6}\right]\), then value of '\(c\)' by applying LMVT is

\(\text { If } \sin (x+y)+\cos (x+y)=\sin \left[\cos ^{-1}\left(\frac{1}{3}\right)\right] \text {, then } \frac{dy}{dx}=\)

The equation of normal to the curve \(y=\sin \left(\frac{\pi x}{4}\right)\) at the point \((2,5)\) is

For every value of \(x\), the function \(f(x)=\frac{1}{a^x}, a>\) 0 is,

If $f(x)=x+\frac{1}{x}, x \neq 0$, then local maximum and minimum values of function $f$ are respectively.......

The slope of normal to the curve $x=\sqrt{t}$ and $y=t-\frac{1}{\sqrt{t}}$ at $t=4$ is ..........

If $r$ is the radius of spherical balloon at time $t$ and the surface area of balloon changes at a constant rate $K$, then......

- The edge of a cube is decreasing at the rate of $0.04 \mathrm{~cm} / \mathrm{sec}$. If the edge of the cube is 10 cms , then rate of decrease of surface area of the cube is...

If $f(x)=3 x^3-9 x^2-27 x+15$, then the maximum value of $f(x)$ is ...........

A stone is dropped into a pond. Waves in the form of circles are generated and radius of outermost ripple increases at the rate of $5 \mathrm{~cm} / \mathrm{sec}$. Then area increased after 2 seconds is ............

The equation of normal to the curve $y=\log _\theta x$ at the point $P(1,0)$ is ............

The function $f(x)=x^3-3 x$ is ............

A particle moves so that $x=2+27 t-t^3$. The direction of motion reverses after moving a distance of ....... units.

Using differentiation, approximate value of $f(x)=x^2-2 x+1$ at $x=2.99$ is ............