Application of Derivatives — JEE Main PYQs

The number of distinct real roots of $x^4-4x+1=0$ is :

The lengths of the sides of a triangle are $10+x^2$, $10+x^2$ and $20-2x^2$. If for $x=k$, the area of the triangle is maximum, then $3k^2$ is equal to :

Let $f(x)=3^{{\left(x^2-2\right)}^3+4},x\in \mathbf{R}$. Then which of the following statements are true ?
$\mathrm{P}:\mathrm{x}=0$ is a point of local minima of f
$\mathrm{Q}:\mathrm{x}=\sqrt{2}$ is a point of inflection of f
R : ${\mathrm{f}}^{\prime }$ is increasin …

The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :

$f(x)=4{\log }_e(x-1)-2x^2+4x+5,x>1$, which one of the following is NOT correct ?

the tangent at the point $\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ on the curve $\mathrm{y}={\mathrm{x}}^3+3{\mathrm{x}}^2+5$ passes through the origin, then $\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ does NOT lie on the curve :

The sum of absolute maximum and absolute minimum values of the function $f(x)=\left|2x^2+3x-2\right|+\sin x\cos x$ in the interval $[0,1]$ is :

Let $\underset{0\le x\le 2}{\mathrm{Max}}\left\{\frac{9-x^2}{5-x}\right\}=\alpha$ and $\underset{0\le x\le 2}{\mathrm{Min}}\left\{\frac{9-x^2}{5-x}\right\}=\beta$

If …

The number of distinct real roots of the equation $x^7-7x-2=0$ is

Let ${\lambda }^{\ast }$ be the largest value of $\lambda$ for which the function $f_{\lambda }(x)=4\lambda x^3-36\lambda x^2+36x+48$ is increasing for all $x\in \mathbb{R}$. Then $f_{{\lambda }^{\ast }}(1)+f_{{\lambda }^{\ast }}(-1)$ is equal to :

Water is being filled at the rate of $1{\mathrm{cm}}^3/\sec$ in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm . When the height of the water level is 10 cm , the rate ( $\mathrm{in}{\mathrm{cm}}^2/\sec$ ) at which the …

Let $f(x)=\left|(x-1)\left(x^2-2x-3\right)\right|+x-3,x\in R$. If $m$ and $M$ are respectively the number of points of local minimum and local maximum of $f$ in the interval $(0,4)$, then $m+M$ is equal to

Let $\mathrm{f}:\mathrm{R}\to \mathrm{R}$ and $\mathrm{g}:\mathrm{R}\to \mathrm{R}$ be two functions defined by $\mathrm{f}(\mathrm{x})={\log }_{\mathrm{e}}\left({\mathrm{x}}^2+1\right)-{\mathrm{e}}^{-\mathrm{x}}+1$ and $g(x)=\frac{1-2e^{2x}}{e^x}$. Then, for …

A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is $k$ (meter), then $\left(\frac{4}{\pi }+1\right)k$ is …

The function $f(x)=\frac{4x^3-3x^2}{6}-2\sin x+(2x-1)\cos x$ :

If the tangent to the curve $y=x^3$ at the point $P\left(t,t^3\right)$ meets the curve again at $Q$, then the ordinate of the point which divides PQ internally in the ratio $1:2$ is:

The minimum value of $\alpha$ for which the equation $\frac{4}{\sin x}+\frac{1}{1-\sin x}=\alpha$ has at least one solution in $\left(0,\frac{\pi }{2}\right)$ is _____.

The local maximum value of the function $f(x)={\left(\frac{2}{x}\right)}^{x^2},x>0$, is:

Let $a$ and $b$ respectively be the points of local maximum and local minimum of the function $F(x)=2x^3-3x^2-12x$. If $A$ is the total area of the region bounded by $y=f(x)$, the $x$-axis and the lines $x=$ a and $\mathrm{x}=\mathrm{b}$, then 4 A is equal to ______.

Let 'a' be a real number such that the function $f(x)=a{x}^2+6x-15,x\in R$ is increasing in $\left(-\infty ,\frac{3}{4}\right)$ and decreasing in $\left(\frac{3}{4},\infty \right)$. Then the function …

Let $f:R\to R$ be defined as
$f(x)=\{\begin{array}{ll}-55x,& \text{ if }x<-5\\ 2x^3-3x^2-120x,& \text{ if }-5\le x\le 4\\ 2x^3-3x^2-36x-336,& \text{ if }x>4,\end{array}$
Let $A=\{x\in R:f$ is increasing $\}$. Then $A$ is equal to:

For which of the following curves, the line $x+\sqrt{3y}=2\sqrt{3}$ is the tangent at the point $\left(\frac{3\sqrt{3}}{2},\frac{1}{2}\right)$ ?

If $P$ is a point on the parabola $y=x^2+4$ which is closest to the straight line $y=4x-1$, then the co-ordination of $P$ are:

The sum of all the local minimum values of the twice differentiable function $f:R\to R$ defined by $f(x)=x^3-3x^2-\frac{3f^{\prime \prime }(2)}{2}x+f\prime \prime (1)$ is :

If the point on the curve $y^2=6x$, nearest to the point $\left(3,\frac{3}{2}\right)$ is $(\alpha ,\beta )$, then $2(\alpha +\beta )$ is equal to. ______

Let $M$ and $m$ respectively be the maximum and minimum values of the function $f(x)={\tan }^{-1}(\sin x+\cos x)$ in $\left[0,\frac{\pi }{2}\right]$. Then the value of $\tan (M-m)$ is equal to

A box open from top is made from a rectangular sheet of dimension $a\times b$ by cutting squares each of side $x$ from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to:

If Rolle's theorem holds for the function $f(x)=x^3-a{x}^2+bx-4,x\in [1,2]$ with $f^{\prime }\left(\frac{4}{3}\right)=0$, then ordered pair $(a,b)$ is equal to :

If the curves, $\frac{x^2}{a}+\frac{y^2}{b}=1$ and $\frac{x^2}{c}+\frac{y^2}{d}=1$ intersect each other at an angle of $90^{\circ }$, then which of the following relations is TRUE ?

Let $f(x)$ be a polynomial of degree 6 in $x$, in which the coefficient of $x^6$ is unity and it has extrema at $x=-1$ and $x=1$. If ${\lim }_{x\to 0}\frac{f(x)}{x^3}=1$, then $5f(2)$ is equal to ____

Let $f:R\to R$ be defined as
$f(x)=\{\begin{array}{cc}-\frac{4}{3}{x}^3+2x^2+3x& ,x>0\\ 3x{e}^x& ,x\le 0\end{array}$
Then $f$ is increasing function in the interval.

The maximum slope of the curve $y=\frac{1}{2}{x}^4-5x^3+18x^2-19x$ occurs at the point :

A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is:

The number of distinct real roots of the equation $3x^4+4x^3-12x^2+4=0$ is

The triangle of maximum area that can be inscribed in a given circle of radius ' $r$ ' is :

An angle of intersection of the curves, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $x^2+y^2=ab,a>b$ is:

Let $f$ be any continuous function on $[0,2]$ and twice differentiable on $(0,2)$. If $f(0)=0,f(1)=1$ and $f(2)=2$, then

Let $f(x)$ be a cubic polynomial with $f(1)=-10,f(-1)=6$, and has a local minima at $x=1$, and $f^{\prime }(x)$ has a local minima at $x=-1$. Then $f(3)$ is equal to ____.

If 'R' is the least value of 'a' such that the function is increasing on and 'S' is the greatest value of 'a' such that the function is decreasing on , then the value of is ______.

Let $f(x)=3{\sin }^{4}x+10{\sin }^{3}x+6{\sin }^{2}x-3,x\in \left[-\frac{\pi }{6},\frac{\pi }{2}\right]$. Then, $f$ is :

If a rectangle is inscribed in an equilateral triangle of side length $2\sqrt{2}$ as shown in the figure, then the square of the largest area of such a rectangle is ____

Let $\alpha ={\max }_{x\in R}\left\{8^{2\sin 3x}\cdot 4^{4\cos 3x}\right\}$ and $\beta ={\min }_{x\in R}\left\{8^{2\sin 3x}\cdot 4^{4\cos 3x}\right\}$. If $8x^2+bx+c=0$ is a quadratic equation whose roots are ${\alpha }^{1/5}$ and ${\beta }^{1/5}$, then the value of $c-b$ …

Let $f:(a,b)\to R$ be twice differentiable such that $f(x)={\int }_a^{x}g(t)dt$ for a differentiable function $g(x)$. If $f(x)=0$ has exactly five distinct roots in $(a,b)$, then $g(x)g^{\prime }(x)=0$ has at least :

The number of real roots of the equation
$e^{4x}-e^{3x}-4e^{2x}-e^x+1=0$ is equal to ____

The function $f(x)=x^3-6x^2+ax+b$ is such that $f(2)=f(4)=0$.
Consider two statements.
(S1) there exists $x_1,x_2\in (2,4),x_1<x_2$, such that $f^{\prime }(x_1)=-1$ and $f^{\prime }(x_2)=0$.
(S2) there exists $x_3,x_4\in (2,4),x_3<x_4$, such that $f$ is decreasing in …

The range of $a\in R$ for which the function $f(x)=(4a-3)\left(x+{\log }_{e}5\right)+2(a-7)\cot \left(\frac{x}{2}\right){\sin }^2\left(\frac{x}{2}\right),x\ne 2n\pi ,n\in N$ has critical points is:

If the normal to the curve $y(x)={\int }_0^x\left(2t^2-15t+10\right)dt$ at a point $(a,b)$ is parallel to the line $x+3y=-5,a>1$, then the value of $|a+6b|$ is equal to ____

Let $f$ be a real valued function, defined on $R-\{-1,1\}$ and given by $f(x)=3{\log }_e\left|\frac{x-1}{x+1}\right|-\frac{2}{x-1}$. Then in which of the following intervals, function $f(x)$ is increasing?

Let $P(x)$ be a real polynomial of degree 3 which vanishes at $x=-3$. Let $P(x)$ have local minima at $x=1$, local maxima at $x=-1$ and ${\int }_{-1}^{1}P(x)dx=18$, then the sum of all the coefficients of the polynomial $P(x)$ is equal to......

Consider the function $f:R\to R$ defined by $f(x)=\{\begin{array}{ll}\left(2-\sin \left(\frac{1}{x}\right)\right)& |x|,x\ne 0\\ 0& ,x=0\end{array}$. Then $f$ is: