Application of Derivatives — JEE Main PYQs

Let $f(x)=2^x-x^2,x\in R$. If $m$ and $n$ are respectively the number of points at which the curves $y=f(x)$ and $y=f^{\prime }(x)$ intersects the $x$-axis, then the value of $m+n$ is

Let $f(x)=(x+3{)}^2(x-2{)}^3,x\in [-4,4]$. If $M$ and $m$ are the maximum and minimum values of f , respectively in $[-4,4]$, then the value of $M-m$ is :

The number of points, where the curve $y=x^5-20x^3+50x+2$ crosses the $x$-axis, is ____.

If $a_n$ is the greatest term in the sequence $a_n=\frac{n^3}{n^4+147},n=1,2,3\dots$, then $n$ is equal to ______.

Let the tangent to the curve $x^2+2x-4y+9=0$ at the point $P(1,3)$ on it meet the y-axis at A. Let the line passing through P and parallel to the line $x-3y=6$ meet the parabola $y^2=4x$ at B. If B lies on the line $2x-3y=8$, then $(AB{)}^2$ is equal to

Let $g(x)=f(x)+f(1-x)$ and $f^{\prime \prime }(x)>0,x\in (0,1)$.
If $g$ is decreasing in the interval $(0,\alpha )$ and increasing in the interval $(\alpha ,1)$, then …

Let quadratic curve passing through the point $(-1,0)$ and touching the line $\mathrm{y}=\mathrm{x}$ at $(1,1)$ be $\mathrm{y}=\mathrm{f}(\mathrm{x})$. Then the x -intercept of the normal to the curve at the point $(\alpha ,\alpha +1)$ in the first quadrant is

A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area ( $\mathrm{in}{\mathrm{cm}}^2$ ) is equal to

If the local maximum value of the function $f(x)={\left(\frac{\sqrt{3e}}{2\sin x}\right)}^{{\sin }^{2}x},x\in \left(0,\frac{\pi }{2}\right)$, is $\frac{k}{e}$, then ${\left(\frac{k}{e}\right)}^8+\frac{k^8}{e^5}+k^8$ is equal to

Consider the triangles with vertices $\mathrm{A}(2,1)$ $\mathrm{B}(0,0)$ and $\mathrm{C}(\mathrm{t},4),\mathrm{t}\in [0,4]$. If the maximum and the minimum perimeters of such triangles are obtained at $\mathrm{t}=\alpha$ and $\mathrm{t}=\beta$ respectively, then …

The number of points, where the curve $f(x)=e^{8x}-e^{6x}-3e^{4x}-e^{2x}+1$, $x\in \mathbb{R}$ cuts $x$-axis, is equal to

The number of points on the curve $y=54x^5-135x^4-70x^3+180x^2+210x$ at which the normal lines are parallel to $x+90y+2=0$ is:

The sum of the abosolute maximum and minimum values of the function $f(x)=\left|x^2-5x+6\right|-3x+2$ in the interval $[-1,3]$ is equal to :

Let f and g be twice differentiable functions on R such that
${\mathrm{f}}^{\prime \prime }(\mathrm{x})={\mathrm{g}}^{\prime \prime }(\mathrm{x})+6\mathrm{x}$
${\mathrm{f}}^{\prime }(1)=4{\mathrm{g}}^{\prime }(1)-3=9$
$\mathrm{f}(2)=3\mathrm{g}(2)=12$
Then which of the following i …

If the equation of the normal to the curve $\mathrm{y}=\frac{\mathrm{x}-\mathrm{a}}{(\mathrm{x}+\mathrm{b})(\mathrm{x}-2)}$ at the point $(1,-3)$ is $\mathrm{x}-4\mathrm{y}=13$, then the value of $\mathrm{a}+\mathrm{b}$ is equal to ______.

Let the function $f(x)=2x^3+(2p-7)x^2+3(2p-9)x-6$ have a maxima for some value of $\mathrm{x}<0$ and a minima for some value of $\mathrm{x}>0$. Then, the set of all values of $p$ is

The absolute minimum value, of the function $\mathrm{f}(\mathrm{x})=\left|{\mathrm{x}}^2-\mathrm{x}+1\right|+\left[{\mathrm{x}}^2-\mathrm{x}+1\right]$, where $[\mathrm{t}]$ denotes the greatest integer function, in the interval $[-1,2]$, is :

If the functions $f(x)=\frac{x^3}{3}+2bx+\frac{a{x}^2}{2}$ and $g(x)=\frac{x^3}{3}+ax+b{x}^2,a\ne 2b$ have a common extreme point, then $\mathrm{a}+2\mathrm{b}+7$ is equal to

The range of the function $\mathrm{f}(\mathrm{x})=\sqrt{3-\mathrm{x}}+\sqrt{2+\mathrm{x}}$ is

A wire of length 20 m is to be cut into two pieces. A piece of length ${\ell }_1$ is bent to make a square of area $A_1$ and the other piece of length ${\ell }_2$ is made into a circle of area $A_2$. If $2A_1+3A_2$ is minimum then $(\pi {\ell }_1):{\ell }_2$ is e …

Let $\mathrm{f}(\mathrm{x})=2\mathrm{x}+{\tan }^{-1}\mathrm{x}$ and $\mathrm{g}(\mathrm{x})={\log }_{\mathrm{e}}\left(\sqrt{1+{\mathrm{x}}^2}+\mathrm{x}\right)$, $x\in [0,3]$. Then

The minimum value of the function $f(x)={\int }_0^2{e}^{|x-t|}dt$ is

Let $\mathrm{x}=2$ be a local minima of the function $f(x)=2x^4-18x^2+8x+12,x\in (-4,4)$. If $M$ is local maximum value of the function f in $(-4,4)$, then $\mathrm{M}=$

Let $f:(0,1)\to \mathbb{R}$ be a function defined by
$\mathrm{f}(\mathrm{x})=\frac{1}{1-{\mathrm{e}}^{-\mathrm{x}}}$, and
$g(x)=(f(-x)-f(x))$. Consider two statements
(I) g is an increasing function in $(0,1)$
(II) g is one-one in $(0,1)$

Then,

If the tangent to the curve $y=x^3-x^2+x$ at the point $(a,b)$ is also tangent to the curve $y=5x^2+2\mathrm{x}-25$ at the point $(2,-1)$, then $|2\mathrm{a}+9\mathrm{b}|$ is equal to ____.

If the minimum value of $f(x)=\frac{5x^2}{2}+\frac{\alpha }{x^5},x>0$, is 14, then the value of $\alpha$ is equal to :

Let $\mathrm{f}:[0,1]\to \mathrm{R}$ be a twice differentiable function in $(0,1)$ such that $\mathrm{f}(0)=3$ and $\mathrm{f}(1)=5$. If the line $\mathrm{y}=2\mathrm{x}+3$ intersects the graph of f at only two distinct points in $(0,1)$, then the least number of points …

The function $\mathrm{f}(\mathrm{x})={\mathrm{xe}}^{\mathrm{x}(1-\mathrm{x})},\mathrm{x}\in \mathrm{R}$, is

The sum of the absolute maximum and absolute minimum values of the function $f(x)={\tan }^{-1}(\sin x-\cos x)$ in the interval $[0,\pi ]$ is

Let $M$ and $N$ be the number of points on the curve $y^5-9xy+2x=0$, where the tangents to the curve are parallel to x-axis and y-axis, respectively. Then the value of $M+N$ equals ______.

If the absolute maximum value of the function $\mathrm{f}(\mathrm{x})=\left({\mathrm{x}}^2-2\mathrm{x}+7\right){\mathrm{e}}^{\left(4{\mathrm{x}}^3-12{\mathrm{x}}^2-180\mathrm{x}+31\right)}$ in the interval [-3, $0]$ is $\mathrm{f}(\alpha )$, then :

The curve $\mathrm{y}(\mathrm{x})={\mathrm{ax}}^3+{\mathrm{bx}}^2+\mathrm{cx}+5$ touches the x -axis at the point $\mathrm{P}(-2,0)$ and cuts the y -axis at the point Q , where ${\mathrm{y}}^{\prime }$ is equal to 3 . Then the local maximum value of $\mathrm{y}(\mathrm{x})$ …

Let P and Q be any points on the curves $(x-1{)}^2+(y+1{)}^2=1$ and $y=x^2$, respectively. The distance between P and Q is minimum for some value of the abscissa of P in the interval

If the maximum value of $a$, for which the function $f_a(x)={\tan }^{-1}2x-3ax+7$ is non-decreasing in $\left(-\frac{\pi }{6},\frac{\pi }{6}\right)$, is $\overline{a}$, then $f_{\overline{a}}\left(\frac{\pi }{8}\right)$ is equal to

Consider a curve $y=y(x)$ in the first quadrant as shown in the figure. Let the area ${\mathrm{A}}_1$ is twice the area ${\mathrm{A}}_2$. Then the normal to the curve perpendicular to the line $2\mathrm{x}-12\mathrm{y}=15$ does NOT pass through the point.

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semivertical angle is ${\tan }^{-1}\frac{3}{4}$. Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface …

The sum of the maximum and minimum values of the function $\mathrm{f}(\mathrm{x})=|5\mathrm{x}-7|+\left[{\mathrm{x}}^2+2\mathrm{x}\right]$ is the interval $\left[\frac{5}{4},2\right]$, where $[t]$ is the greatest integer $\le \mathrm{t}$ is

Let the area enclosed by the x-axis, and the tangent and normal drawn to the curve $4x^3-3x{y}^2+6x^2-5xy-8y^2+9x+14=0$ at the point $(-2,3)$ be A. Then 8A is equal to

Let $f(x)=\{\begin{array}{cc}{x}^3-x^2+10x-7,& x\le 1\\ -2x+{\log }_2\left(b^2-4\right),& x>1\end{array}$

Then the set of all values of $b$, for which $f(x)$ has maximum value at $\mathrm{x}=1$, is :

Let the function $f(x)=2x^2-{\log }_{e}x,x>0$, be decreasing in $(0,a)$ and increasing in $(a,4)$. A tangent to the parabola $y^2=4ax$ at a point P on it passes through the point $(8a,8a-1)$ but does not pass through the point $\left(-\frac{1}{a},0\right)$. If the equation …

If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $f(x)={\int }_0^{x^2}\frac{t^2-5t+4}{2+e^t}dt$, then the ordered pair $(m,n)$ is equal to

The number of real solutions of $x^7+5x^3+3x+1=0$ is equal to

Let $\ell$ be a line which is normal to the curve $y=2x^2+x+2$ at a point P on the curve. If the point $Q(6,4)$ lies on the line $\ell$ and O is origin, then the area of the triangle OPQ is equal to ____.

Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined by $f(x)=(x-3{)}^{n_1}(x-5{)}^{n_2},n_1,n_2\in \mathbb{N}$. Then, which of the following is NOT true?

Let $f$ and g be twice differentiable even functions on $(-2,2)$ such that $f\left(\frac{1}{4}\right)=0,f\left(\frac{1}{2}\right)=0,f(1)=1$ and $g\left(\frac{3}{4}\right)=0,g(1)=2$. Then, the minimum number of solutions of …

Consider a cuboid of sides 2 x , 4 x and 5 x and a closed hemisphere of radius r . If the sum of their surface areas is a constant k , then the ratio $\mathrm{x}:\mathrm{r}$, for which the sum of their volumes is maximum, is :

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, …

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

Let S be the set of all the natural numbers, for which the line $\frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=2$ is a tangent to the curve ${\left(\frac{\mathrm{x}}{\mathrm{a}}\right)}^{\mathrm{n}}+{\left(\frac{\mathrm{y}}{\mathrm{b}}\right)}^{\mathrm{n}}=2$ at the …