Application of Derivatives
584 Questions
Start JEE Mains Test
1993
Q551
JEE Advanced
Numerical
14 Mar 2026
Let $f\left( x \right) = \left\{ {\matrix{
{ - {x^3} + {{\left( {{b^3} - {b^2} + b - 1} \right)} \over {\left( {{b^2} + 3b + 2} \right)}},} & {0 \le x < 1} \cr
{2x - 3} & {1 \le x \le 3} \cr
} } \right.$
Find all possible real values of $b$ such that $f(x)$ has the smallest value at $x=1$.
Correct Answer: $$b \in \left( { - 2, - 1} \right) \cup \left( {1,x} \right)$$
1993
Q552
JEE Advanced
Numerical
14 Mar 2026
Find the equation of the normal to the curve
$y = {\left( {1 + x} \right)^y} + {\sin ^{ - 1}}\left( {{{\sin }^2}x} \right)$ at $x=0$
$y = {\left( {1 + x} \right)^y} + {\sin ^{ - 1}}\left( {{{\sin }^2}x} \right)$ at $x=0$
Correct Answer: $$x+y=1$$
1992
Q553
JEE Advanced
Numerical
14 Mar 2026
What normal to the curve $y = {x^2}$ forms the shortest chord?
Correct Answer: <p>$$x + \sqrt 2 \,y = \sqrt 2 $$</p>
<p>or</p>
<p>$$x - \sqrt {2\,} y = - \sqrt 2 $$</p>
1992
Q554
JEE Advanced
Numerical
14 Mar 2026
In this questions there are entries in columns $I$ and $II$. Each entry in column $I$ is related to exactly one entry in column $II$. Write the correct letter from column $II$ against the entry number in column $I$ in your answer book.
Let the functions defined in column $I$ have domain $\left( { - {\pi \over 2},{\pi \over 2}} \right)$
$\,\,\,\,$Column $I$
(A) $x + \sin x$
(B) $\sec x$
$\,\,\,\,$Column $II$
(p) increasing
(q) decreasing
(r) neither increasing nor decreasing
Correct Answer: $$\left( A \right) - \left( p \right),\left( B \right) - \left( r \right)$$
1992
Q555
JEE Advanced
Numerical
14 Mar 2026
A cubic $f(x)$ vanishes at $x=2$ and has relative minimum / maximum at $x=-1$ and $x = {1 \over 3}$ if $\int\limits_{ - 1}^1 {f\,\,dx = {{14} \over 3}} $, find the cubic $f(x)$.
Correct Answer: $${x^3} + {x^2} - x + 2$$
1991
Q556
JEE Advanced
Numerical
14 Mar 2026
A window of perimeter $P$ (including the base of the arch) is in the form of a rectangle surmounded by a semi circle. The semi-circular portion is fitted with coloured glass while the rectangular part is fitted with clear glass transmits three times as such light per square meter as the coloured glass does.
What is the ratio for the sides of the rectangle so that the window transmits the maximum light ?
Correct Answer: $${{6 + \pi } \over 6}$$
1990
Q557
JEE Advanced
Numerical
14 Mar 2026
Show that $2\sin x + \tan x \ge 3x$ where $0 \le x < {\pi \over 2}$.
Correct Answer: Solve it.
1990
Q558
JEE Advanced
Numerical
14 Mar 2026
A point $P$ is given on the circumference of a circle of radius $r$. Chord $QR$ is parallel to the tangent at $P$. Determine the maximum possible area of the triangle $PQR$.
Correct Answer: $${{3\sqrt 3 } \over 4}\,\,{r^2}$$
1989
Q559
JEE Advanced
Numerical
14 Mar 2026
Find all maxima and minima of the function
$$y = x{\left( {x - 1} \right)^2},0 \le x \le 2$$
Also determine the area bounded by the curve $y = x{\left( {x - 1} \right)^2}$,
the $y$-axis and the line $y-2$.
Also determine the area bounded by the curve $y = x{\left( {x - 1} \right)^2}$,
the $y$-axis and the line $y-2$.
Correct Answer: $${{10 \over 3}}$$ sq. units
1988
Q560
JEE Advanced
Numerical
14 Mar 2026
Investigate for maxima and minimum the function
$$f\left( x \right) = \int\limits_1^x {\left[ {2\left( {t - 1} \right){{\left( {t - 2} \right)}^3} + 3{{\left( {t - 1} \right)}^2}{{\left( {t - 2} \right)}^2}} \right]} dt$$
Correct Answer: min at $$x = {7 \over 5}$$
<br>max at $$x = 1$$.
1987
Q561
JEE Advanced
MCQ
14 Mar 2026
Let $f$ and $g$ be increasing and decreasing functions, respectively from $\left[ {0,\infty } \right)$ to $\left[ {0,\infty } \right)$. Let $h\left( x \right) = f\left( {g\left( x \right)} \right).$ If $h\left( 0 \right) = 0,$ then $h\left( x \right) - h\left( 1 \right)$ is
A.
always zero
B.
always negative
C.
always positive
D.
strictly increasing
1987
Q562
JEE Advanced
MCQ
14 Mar 2026
The smallest positive root of the equation, $\tan x - x = 0$ lies in
A.
$\left( {0,{\pi \over 2}} \right)$
B.
$\left( {{\pi \over 2},\pi } \right)$
C.
$\left( {\pi ,{{3\pi } \over 2}} \right)$
D.
$\left( {{{3\pi } \over 2},2\pi } \right)$
1987
Q563
JEE Advanced
Numerical
14 Mar 2026
Find the point on the curve $\,\,\,4{x^2} + {a^2}{y^2} = 4{a^2},\,\,\,4 < {a^2} < 8$
that is farthest from the point $(0, -2)$.
that is farthest from the point $(0, -2)$.
Correct Answer: $$(0, 2)$$
1987
Q564
JEE Advanced
Numerical
14 Mar 2026
The set of all $x$ for which $in\left( {1 + x} \right) \le x$ is equal to ..........
Correct Answer: $$x \ge 0$$
1986
Q565
JEE Advanced
MCQ
14 Mar 2026
Let $P\left( x \right) = {a_0} + {a_1}{x^2} + {a_2}{x^4} + ...... + {a_n}{x^{2n}}$ be a polynomial in a real variable $x$ with
$0 < {a_0} < {a_1} < {a_2} < ..... < {a_n}.$ The function $P(x)$ has
$0 < {a_0} < {a_1} < {a_2} < ..... < {a_n}.$ The function $P(x)$ has
A.
neither a maximum nor a minimum
B.
only one maximum
C.
only one minimum
D.
only one maximum and only one minimum
1986
Q566
JEE Advanced
MSQ
14 Mar 2026
If the line $ax+by+c=0$ is a normal to the curve $xy=1$, then
A.
$a > 0,b > 0$
B.
$a > 0,b < 0$
C.
$a < 0,b > 0$
D.
$a < 0,b < 0$
1985
Q567
JEE Advanced
Numerical
14 Mar 2026
Find all the tangents to the curve
$y = \cos \left( {x + y} \right),\,\, - 2\pi \le x \le 2\pi ,$ that are parallel to the line $x+2y=0$.
$y = \cos \left( {x + y} \right),\,\, - 2\pi \le x \le 2\pi ,$ that are parallel to the line $x+2y=0$.
Correct Answer: $$2x + 4y - \pi = 0$$
<br>$$2x + 4y + 3\pi = 0$$
1985
Q568
JEE Advanced
Numerical
14 Mar 2026
Let $f\left( x \right) = {\sin ^3}x + \lambda {\sin ^2}x, - {\pi \over 2} < x < {\pi \over 2}.$ Find the intervals in which $\lambda $ should lie in order that $f(x)$ has exactly one minimum and exactly one maximum.
Correct Answer: $$\lambda \in \left( { - {3 \over 2},0} \right) \cup \left( {0,{3 \over 2}} \right)$$
1984
Q569
JEE Advanced
MCQ
14 Mar 2026
For $0 < a < x,$ the minimum value of the function $lo{g_a}x + {\log _x}a$ is $2$.
A.
TRUE
B.
FALSE
1983
Q570
JEE Advanced
MCQ
14 Mar 2026
The normal to the curve $\,x = a\left( {\cos \theta + \theta \sin \theta } \right)$, $y = a\left( {\sin \theta - \theta \cos \theta } \right)$ at any point $'\theta '$ is such that
A.
it makes a constant angle with the $x$-axis
B.
it passes through the origin
C.
it is at a constant distance from the origin
D.
none of these
1983
Q571
JEE Advanced
MCQ
14 Mar 2026
If $a+b+c=0$, then the quadratic equation $3a{x^2} + 2bx + c = 0$ has
A.
at least one root in $\left[ {0,1} \right]$
B.
one root in $\left[ {2,3} \right]$ and the other in $\left[ {-2,-1} \right]$
C.
imaginary roots
D.
none of these
1983
Q572
JEE Advanced
MCQ
14 Mar 2026
$AB$ is a diameter of a circle and $C$ is any point on the circumference of the circle. Then
A.
the area of $\Delta ABC$ is maximum when it is isosceles
B.
the area of $\Delta ABC$ is minimum when it is isosceles
C.
the perimeter of $\Delta ABC$ is minimum when it is isosceles
D.
none of these
1983
Q573
JEE Advanced
MCQ
14 Mar 2026
If $y = a\,\,In\,x + b{x^2} + x$ has its extreamum values at $x=-1$ and $x=2$, then
A.
$a = 2,b = - 1$
B.
$a = 2,b = - {1 \over 2}$
C.
$a = - 2,b = {1 \over 2}$
D.
none of these
1983
Q574
JEE Advanced
Numerical
14 Mar 2026
Show that $1+x$ $In\left( {x + \sqrt {{x^2} + 1} } \right) \ge \sqrt {1 + {x^2}} $ for all $x \ge 0$
Correct Answer: Solve it.
1983
Q575
JEE Advanced
Numerical
14 Mar 2026
Find the coordinates of the point on the curve $y = {x \over {1 + {x^2}}}$
where the tangent to the curve has the greatest slope.
where the tangent to the curve has the greatest slope.
Correct Answer: $$(0. 0)$$
1983
Q576
JEE Advanced
Numerical
14 Mar 2026
The larger of $\cos \left( {In\,\,\theta } \right)$ and $In $ $\left( {\cos \,\,\theta } \right)$ If ${e^{ - \pi /2}} < \theta < {\pi \over 2}$ is ..................
Correct Answer: $$\cos \left( {In\,\,\theta } \right)$$
1983
Q577
JEE Advanced
Numerical
14 Mar 2026
The function $y = 2{x^2} - In\,\left| x \right|$ is monotonically increasing for values of $x\left( {x \ne 0} \right)$ satisfying the inequalities ......... and monotonically decreasing for values of $x$ satisfying the inequalities ............
Correct Answer: $$x \in \left( { - {1 \over 2},{\mkern 1mu} 0} \right) \cup \left( {{1 \over 2},{\mkern 1mu} \alpha } \right);{\mkern 1mu} {\mkern 1mu} \left( { - \alpha ,{\mkern 1mu} - {1 \over 2}} \right) \cup \left( {0,{\mkern 1mu} {1 \over 2}} \right)$$
1983
Q578
JEE Advanced
MCQ
14 Mar 2026
If $x-r$ is a factor of the polynomial $f\left( x \right) = {a_n}{x^4} + ..... + {a_0},$ repeated $m$ times $\left( {1 < m \le n} \right)$, then $r$ is a root of $\left( x \right) = 0$ repeated $m$ times.
A.
TRUE
B.
FALSE
1982
Q579
JEE Advanced
Numerical
14 Mar 2026
If $a{x^2} + {b \over x} \ge c$ for all positive $x$ where $a>0$ and $b>0$ show that $27a{b^2} \ge 4{c^3}$.
Correct Answer: Solve it.
1982
Q580
JEE Advanced
Numerical
14 Mar 2026
If $f(x)$ and $g(x)$ are differentiable function for $0 \le x \le 1$ such that $f(0)=2$, $g(0)=0$, $f(1)=6$; $g(1)=2$, then show that there exist $c$ satisfying $0 < c < 1$ and $f'(c)=2g'(c)$.
Correct Answer: Solve it.
1981
Q581
JEE Advanced
Numerical
14 Mar 2026
Use the function $f\left( x \right) = {x^{1/x}},x > 0$. to determine the bigger of the two numbers ${e^\pi }$ and ${\pi ^e}$
Correct Answer: $${e^\pi }$$
1981
Q582
JEE Advanced
Numerical
14 Mar 2026
Let $x$ and $y$ be two real variables such that $x>0$ and $xy=1$. Find the minimum value of $x+y$.
Correct Answer: $$2$$
1981
Q583
JEE Advanced
Numerical
14 Mar 2026
For all $x$ in $\left[ {0,1} \right]$, let the second derivative $f''(x)$ of a function $f(x)$ exist and satisfy $\left| {f''\left( x \right)} \right| < 1.$ If $f(0)=f(1)$, then show that $\left| {f\left( x \right)} \right| < 1$ for all $x$ in $\left[ {0,1} \right]$.
Correct Answer: Solve it.
1979
Q584
JEE Advanced
Numerical
14 Mar 2026
Prove that the minimum value of ${{\left( {a + x} \right)\left( {b + x} \right)} \over {\left( {c + x} \right)}},$
$a,b > c,x > - c$ is ${\left( {\sqrt {a - c} + \sqrt {b - c} } \right)^2}$.
$a,b > c,x > - c$ is ${\left( {\sqrt {a - c} + \sqrt {b - c} } \right)^2}$.
Correct Answer: Solve it