Application of Derivatives

If $f(x)=x e^{x(1-x)}, x \in R$, then $f(x)$ is

The angle between the curves $y^2=x$ and $x^2=y$ at the point $(1,1)$ is

The difference between the absolute maximum and absolute minimum values of the function $f(x)=2 x^3-15 x^2+36 x-30$ on $[-1,4]$ is

Consider the quadratic equation $a x^2+b x+c=0$, where $2 a+3 b+6 c=0$ and let $g(x)=\frac{a x^3}{3}+\frac{b x^2}{2}+c x$
Statement I The given quadratic equation $a x^2+b x+c=0$ has atleast one root in $(0,1)$.
Statement II Rolle's theorem...

If the function $f(x)=\sin x-\cos ^2 x$ is defined on the interval $[-\pi, \pi]$, then $f$ is strictly increasing in the interval

If the tangent of the curve $4 y^3=3 a x^2+x^3$ drawn at the point $(a, a)$ forms a triangle of area $\frac{25}{24}$ sq. units with the coordinates axes, then $a=$

If the Lagrange' mean value theorem is applied to the function $f(x)=e^x$ defined on the interval $[1,2]$ and the value of $c \in(1,2)$ is $k$, then $e^{k-1}=$

If the displacement $S$ of a particle travelling along a straight line in $t$ seconds is given by $S=2 t^3+2 t^2-2 t-3$, then the time taken (in second) by the particle to change its direction is

If the function $f(x)=x^3+b x^2+c x-6$ satisfies all the conditions of Rolle's theorem in $[1,3]$ and $f^{\prime}\left(\frac{2 \sqrt{3}+1}{\sqrt{3}}\right)=0$, then $b c=$

If the tangent of the curve $x y+a x+b y=0$ at $(1,1)$ makes an angle $\tan ^{-1} 2$ with $X$-axis, then $\frac{a b}{a+b}=$

The number of turning points of the curve $f(x)=2 \cos x-\sin 2 x$ in the interval $[-\pi, \pi]$ is

The radius and the height of a right circular solid cone are measured as 7 feet each. If there is an error of 0.002 ft for every feet in measuring them, then the error in the total surface area of the cone (in sq. ft ) is

If the surface area of a spherical bubble is increasing at the rate of $4 \mathrm{sq} . \mathrm{cm} / \mathrm{sec}$, then the rate of change in its volume (in cubic $\mathrm{cm} / \mathrm{sec}$ ) when its radius is 8 cms is

Let $P(x)=x^4+a x^3+b x^2+c x+d$ be such that $x=0$ is the only real root of $P^1(x)=0$. If $P(-1)

$P$ and $Q$ are the ends of a diameter of the circle $x^2+y^2=a^2\left(a>\frac{1}{\sqrt{2}}\right) . s$ and $t$ are the lengths of the perpendiculars drawn from $P$ and $Q$ onto the line $x+y=1$ respectively. When the product st is maximum,...

If $\frac{1}{2} \leq \frac{x^2+x+a}{x^2-x+a} \leq 2 \forall x \in R$, then $a=$

If the volume of a sphere is increasing at the rate of 12 c.c. $/ \mathrm{sec}$, then the rate (in $\mathrm{sq} . \mathrm{cm} / \mathrm{sec}$ ) at which its surface area is increasing, when the diameter of the sphere is 12 cm is

If the lengths of the tangent, subtangent, normal and subnormal for the curve $y=x^2+x-1$ at the point $(1,1)$ are $a, b, c$ and $d$ respectively, then their increasing order is

If $m$ and $M$ are the absolute minimum and absolute maximum values of the function $f(x)=2 \sqrt{2} \sin x-\tan x$ in the interval $[0, \pi / 3]$, then $m+M=$

The function $f(x)=x e^{-x} \forall x \in R$ attains a maximum value at $x=k$, then $k=$

The slope of a tangent drawn at the point $P(\alpha, \beta)$ lying on the curve $y=\frac{1}{2 x-5}$ is -2 . If $P$ lies in the fourth quadrant, then $\alpha-\beta=$

If $y=|\cos x-\sin x|+|\tan x-\cot x|$, then

\(\left(\frac{d y}{d x}\right)_{x=\frac{\pi}{3}}+\left(\frac{d y}{d x}\right)_{x=\frac{\pi}{6}}=\)

If the extreme value of the function $f(x)=\frac{4}{\sin x}+\frac{1}{1-\sin x}$ in $\left[0, \frac{\pi}{2}\right]$ is $m$ and it exists at $x=k$, then $\cos k=$

The displacement $S$ of a particle measured from a fixed point $O$ on a line is given by $S=t^3-16 t^2+64 t-16$. Then, the time at which displacement of the particle is maximum is

If the tangent drawn at the point $(\alpha, \beta)$ on the curve $x^{\frac{2}{3}}+y^{\frac{2}{3}}=4$ is parallel to the line $\sqrt{3 x}+y=1$, then $\alpha^2+\beta^2=$

If the area of a right-angle triangle with hypotenuse 5 is maximum, then its perimeter is

If $\beta$ is an angle between the normals drawn to the curve $x^2+3 y^2=9$ at the points $(3 \cos \theta, \sqrt{3} \sin \theta)$ and $(-3 \sin \theta, \sqrt{3} \cos \theta), \theta \in\left(0, \frac{\pi}{2}\right)$, then

Which one of the following functions is monotonically increasing in its domain?

If the tangent drawn at the point $\left(x_1, y_1\right), x_1, y_1 \in N$ on the curve $y=x^4-2 x^3+x^2+5 x$ passes through origin, then $x_1+y_1=$

If the velocity of a particle moving on a straight line is proportional to the cube root of its displacement, then its acceleration is

If the function $y=\sin x(1+\cos x)$ is defined in the interval $[-\pi, \pi]$, then $y$ is strictly increasing in the interval

If the normal drawn at the point $P$ on the curve $y=x \log x$ is parallel to the line $2 x-2 y=3$, then $P=$

If the curves $y^2=16 x$ and $9 x^2+\alpha y^2=25$ intersect at right angles, then $\alpha=$

The area (in square units) of the triangle formed by the $X$-axis, the tangent and the normal drawn at $(1,1)$ to the curve $x^3+y^3=2 x y$ is

If the function $y=g(x)$ representing the slopes of the tangents drawn to the curve $y=3 x^4-5 x^3-12 x^2+18 x+3$ is strictly increasing, then the domain of $g(x)$ is

If $\alpha$ and $\beta(\alpha>\beta)$ are the multiple roots of the equation $4 x^4+4 x^3-23 x^2-12 x+36=0$, then $2 \alpha-\beta=$

The value of the Rolle's theorem for the function $f(x)=2 \sin x+\sin 2 x$ in the interval $[0, \pi]$ is

$A$ is a point on the circle with radius 8 and centre at $O$. A particle $P$ is moving on the circumference of the circle starting from $A . M$ is the foot of the perpendicular from $P$ on $O A$ and $\angle P O M=\theta$. When $O M$ $=4$ an...

If the length of the sub-tangent at any $P$ on a curve is proportional to the abscissa of the point $P$, then the equation of that curve is ( $C$ is an arbitrary constant)

If a running track of 500 ft is to be laid out enclosing a playground the shape of which is a rectangle with a semi-circle at each end, then the length of the rectangular portion such that the area of the rectangular portion is to be maximu...

Displacement $s$ of a particle at time $t$ is expressed as $s=2 t^3-9 t$. Find the acceleration at the time when $b^{t 5}$ velocity vanishes.

Equation of the normal to the curve $y=x^2+x$ at the point $(1,2)$ is

The value of Lagrange's mean value theorem for $f(x)=e^x+24$ in $[0,1]$ is

The semi-vertical angle of a right circular cone is $45^{\circ} \%$ If the radius of the base of the cone is measured as 14 cm with an error of $\left(\frac{\sqrt{2}-1}{11}\right) \mathrm{cm}$, then the approximate error in measuring its to...

The interval containing all the real values of $x$ such that the real valued function $f(x)=\sqrt{x}+\frac{1}{\sqrt{x}}$ is strictly increasing is

If a man of height 1.8 mt , is walking away from the foot of a light pole of height 6 mt , with a speed of 7 km per hour on a straight horizontal road opposite to the pole, then the rate of change of the length of his shadow is (in kmph )

If the curves $2 x^2+k y^2=30$ and $3 y^2=28 x$ cut each other orthogonally, then $k$ is equal to

If a number is drawn at random from the set $\{1,3,5,7, \ldots . .59\}$, then the probability that it lies in the interval in which the function $f(x)=x^3-16 x^2+20 x-5$ is stricly decreasing is

The equation of the normal drawn to the parabola $y^2=6 x$ at the point $(24,12)$ is

The number of all the value of $x$ for which the function $f(x)=\sin x+\frac{1-\tan ^2 x}{1+\tan ^2 x}$ attains it maximum value on [ $0.2 \pi$ ] is