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Application of Derivatives

127 Questions Start JEE Advanced Test
All Exams JEE Mains JEE Advanced TS-EAMCET AP-EAPCET BITSAT
2004 Q51 JEE Advanced Numerical
14 Mar 2026
Prove that for $x \in \left[ {0,{\pi \over 2}} \right],$ $\sin x + 2x \ge {{3x\left( {x + 1} \right)} \over \pi }$. Explain
the identity if any used in the proof.
2004 Q52 JEE Advanced Numerical
14 Mar 2026
Using Rolle's theorem, prove that there is at least one root
in $\left( {{{45}^{1/100}},46} \right)$ of the polynomial
$P\left( x \right) = 51{x^{101}} - 2323{\left( x \right)^{100}} - 45x + 1035$.
2003 Q53 JEE Advanced MCQ
14 Mar 2026
In $\left[ {0,1} \right]$ Languages Mean Value theorem is NOT applicable to
A.
$f\left( x \right) = \left\{ {\matrix{ {{1 \over 2} - x} & {x < {1 \over 2}} \cr {{{\left( {{1 \over 2} - x} \right)}^2}} & {x \ge {1 \over 2}} \cr } } \right.$
B.
$f\left( x \right) = \left\{ {\matrix{ {\sin x,} & {x \ne 0} \cr {1,} & {x = 0} \cr } } \right.$
C.
$f\left( x \right) = x\left| x \right|$
D.
$f\left( x \right) = \left| x \right|$
2003 Q54 JEE Advanced MCQ
14 Mar 2026
Tangent is drawn to ellipse
${{{x^2}} \over {27}} + {y^2} = 1\,\,\,at\,\left( {3\sqrt 3 \cos \theta ,\sin \theta } \right)\left( {where\,\,\theta \in \left( {0,\pi /2} \right)} \right)$.

Then the value of $\theta $ such that sum of intercepts on axes made by this tangent is minimum, is

A.
$\pi /3$
B.
$\pi /6$
C.
$\pi /8$
D.
$\pi /4$
2003 Q55 JEE Advanced Numerical
14 Mar 2026
Find a point on the curve ${x^2} + 2{y^2} = 6$ whose distance from
the line $x+y=7$, is minimum.
2003 Q56 JEE Advanced Numerical
14 Mar 2026
Using the relation $2\left( {1 - \cos x} \right) < {x^2},\,x \ne 0$ or otherwise,
prove that $\sin \left( {\tan x} \right) \ge x,\,\forall x \in \left[ {0,{\pi \over 4}} \right]$
2003 Q57 JEE Advanced Numerical
14 Mar 2026
If the function $f:\left[ {0,4} \right] \to R$ is differentiable then show that
(i)$\,\,\,\,\,$ For $a, b$$\,\,$$ \in \left( {0,4} \right),{\left( {f\left( 4 \right)} \right)^2} - {\left( {f\left( 0 \right)} \right)^2} = gf'\left( a \right)f\left( b \right)$
(ii)$\,\,\,\,\,$ $\int\limits_0^4 {f\left( t \right)dt = 2\left[ {\alpha f\left( {{\alpha ^2}} \right) + \beta \left( {{\beta ^2}} \right)} \right]\forall 0 < \alpha ,\beta < 2} $
2003 Q58 JEE Advanced Numerical
14 Mar 2026
If $P(1)=0$ and ${{dp\left( x \right)} \over {dx}} > P\left( x \right)$ for all $x \ge 1$ then prove that
$P(x)>0$ for all $x>1$.
2002 Q59 JEE Advanced MCQ
14 Mar 2026
The length of a longest interval in which the function $3\,\sin x - 4{\sin ^3}x$ is increasing, is
A.
${\pi \over 3}$
B.
${\pi \over 2}$
C.
${3\pi \over 2}$
D.
$\pi $
2002 Q60 JEE Advanced MCQ
14 Mar 2026
The point(s) in the curve ${y^3} + 3{x^2} = 12y$ where the tangent is vertical, is (are)
A.
$\left( { \pm {4 \over {\sqrt 3 }}, - 2} \right)$
B.
$\left( { \pm \sqrt {{{11} \over 3}} ,1} \right)$
C.
$(0,0)$
D.
$\left( { \pm {4 \over {\sqrt 3 }}, 2} \right)$
2001 Q61 JEE Advanced MCQ
14 Mar 2026
Let $f\left( x \right) = \left( {1 + {b^2}} \right){x^2} + 2bx + 1$ and let $m(b)$ be the minimum value of $f(x)$. As $b$ varies, the range of $m(b)$ is
A.
$\left[ {0,1} \right]$
B.
$\left( {0,\,1/2} \right]$
C.
$\left[ {1/2,\,1} \right]$
D.
$\left( {0,\,1} \right]$
2001 Q62 JEE Advanced MCQ
14 Mar 2026
The triangle formed by the tangent to the curve $f\left( x \right) = {x^2} + bx - b$ at the point $(1, 1)$ and the coordinate axex, lies in the first quadrant. If its area is $2$, then the value of $b$ is
A.
$-1$
B.
$3$
C.
$-3$
D.
$1$
2001 Q63 JEE Advanced MCQ
14 Mar 2026
If $f\left( x \right) = x{e^{x\left( {1 - x} \right)}},$ then $f(x)$ is
A.
increasing on $\left[ { - 1/2,1} \right]$
B.
decreasing on $R$
C.
increasing on $R$
D.
decreasing on $\left[ { - 1/2,1} \right]$
2001 Q64 JEE Advanced Numerical
14 Mar 2026
Let $ - 1 \le p \le 1$. Show that the equation $4{x^3} - 3x - p = 0$
has a unique root in the interval $\left[ {1/2,\,1} \right]$ and identify it.
2000 Q65 JEE Advanced MCQ
14 Mar 2026
Consider the following statements in $S$ and $R$
$S:$ $\,\,\,$$ Both $\sin \,\,x$ and $\cos \,\,x$ are decreasing functions in the interval $\left( {{\pi \over 2},\pi } \right)$
$R:$$\,\,\,$ If a differentiable function decreases in an interval $(a, b)$, then its derivative also decreases in $(a, b)$.
Which of the following is true ?
A.
Both $S$ and $R$ are wrong
B.
Both $S$ and $R$ are correct, but $R$ is not the correct explanation of $S$
C.
$S$ is correct and $R$ is the correct explanation for $S$
D.
$S$ is correct and $R$ is wrong
2000 Q66 JEE Advanced MCQ
14 Mar 2026
Let $f\left( x \right) = \int {{e^x}\left( {x - 1} \right)\left( {x - 2} \right)dx.} $ Then $f$ decreases in the interval
A.
$\left( { - \infty ,2} \right)$
B.
$\left( { - 2, - 1} \right)$
C.
$\left( {1,2} \right)$
D.
$\left( {2, + \infty } \right)$
2000 Q67 JEE Advanced MCQ
14 Mar 2026
Let $f\left( x \right) = \left\{ {\matrix{ {\left| x \right|,} & {for} & {0 < \left| x \right| \le 2} \cr {1,} & {for} & {x = 0} \cr } } \right.$ then at $x=0$, $f$ has
A.
a local maximum
B.
no local maximum
C.
a local minimum
D.
no extremum
2000 Q68 JEE Advanced MCQ
14 Mar 2026
If the normal to the curve $y = f\left( x \right)$ and the point $(3, 4)$ makes an angle ${{{3\pi } \over 4}}$ with the positive $x$-axis, then $f'\left( 3 \right) = $
A.
$-1$
B.
$ - {3 \over 4}$
C.
${4 \over 3}$
D.
$1$
2000 Q69 JEE Advanced MCQ
14 Mar 2026
For all $x \in \left( {0,1} \right)$
A.
${e^x} < 1 + x$
B.
${\log _e}\left( {1 + x} \right) < x$
C.
$\sin x > x$
D.
${\log _e}x > x$
2000 Q70 JEE Advanced Numerical
14 Mar 2026
Suppose $p\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + .......... + {a_n}{x^n}.$ If
$\left| {p\left( x \right)} \right| \le \left| {{e^{x - 1}} - 1} \right|$ for all $x \ge 0$, prove that
$\left| {{a_1} + 2{a_2} + ........ + n{a_n}} \right| \le 1$.
1999 Q71 JEE Advanced MCQ
14 Mar 2026
The function $f(x)=$ ${\sin ^4}x + {\cos ^4}x$ increases if
A.
$0 < x < \pi /8$
B.
$\pi /4 < x < 3\pi /8$
C.
$3\pi /8 < x < 5\pi /8$
D.
$5\pi /8 < x < 3\pi /4$
1999 Q72 JEE Advanced MSQ
14 Mar 2026
The function $f\left( x \right) = \int\limits_{ - 1}^x {t\left( {{e^t} - 1} \right)\left( {t - 1} \right){{\left( {t - 2} \right)}^3}\,\,\,{{\left( {t - 3} \right)}^5}} $ $dt$ has a local minimum at $x=$
A.
$0$
B.
$1$
C.
$2$
D.
$3$
1998 Q73 JEE Advanced MCQ
14 Mar 2026
If $f\left( x \right) = {{{x^2} - 1} \over {{x^2} + 1}},$ for every real number $x$, then the minimum value of $f$
A.
does not exist because $f$ is unbounded
B.
is not attained even though $f$ is bounded
C.
is equal to 1
D.
is equal to -1
1998 Q74 JEE Advanced MCQ
14 Mar 2026
The number of values of $x$ where the function
$f\left( x \right) = \cos x + \cos \left( {\sqrt 2 x} \right)$ attains its maximum is
A.
$0$
B.
$1$
C.
$2$
D.
infinite
1998 Q75 JEE Advanced MSQ
14 Mar 2026
Let $h\left( x \right) = f\left( x \right) - {\left( {f\left( x \right)} \right)^2} + {\left( {f\left( x \right)} \right)^3}$ for every real number $x$. Then
A.
$h$ is increasing whenever $f$ is increasing
B.
$h$ is increasing whenever $f$ is decreasing
C.
$h$ is decreasing whenever $f$ is decreasing
D.
nothing can be said in general.
1998 Q76 JEE Advanced Numerical
14 Mar 2026
Suppose $f(x)$ is a function satisfying the following conditions
(a) $f(0)=2,f(1)=1$,
(b) $f$has a minimum value at $x=5/2$, and
(c) for all $x$, $$f'\left( x \right) = \matrix{ {2ax} & {2ax - 1} & {2ax + b + 1} \cr b & {b + 1} & { - 1} \cr {2\left( {ax + b} \right)} & {2ax + 2b + 1} & {2ax + b} \cr } $$
where $a,b$ are some constants. Determine the constants $a, b$ and the function $f(x)$.
1998 Q77 JEE Advanced Numerical
14 Mar 2026
A curve $C$ has the property that if the tangent drawn at any point $P$ on $C$ meets the co-ordinate axes at $A$ and $B$, then $P$ is the mid-point of $AB$. The curve passes through the point $(1, 1)$. Determine the equation of the curve.
1997 Q78 JEE Advanced MCQ
14 Mar 2026
If $f\left( x \right) = {x \over {\sin x}}$ and $g\left( x \right) = {x \over {\tan x}}$, where $0 < x \le 1$, then in this interval
A.
both $f(x)$ and $g(x)$ are increasing functions
B.
both $f(x)$ and $g(x)$ are decreasing functions
C.
$f(x)$ is an increasing functions
D.
$g(x)$ is an increasing functions
1997 Q79 JEE Advanced Numerical
14 Mar 2026
Let $a+b=4$, where $a<2,$ and let $g(x)$ be a differentiable function.

If ${{dg} \over {dx}} > 0$ for all $x$, prove that $\int_0^a {g\left( x \right)dx + \int_0^b {g\left( x \right)dx} } $
increases as $(b-a)$ increases.

1996 Q80 JEE Advanced Numerical
14 Mar 2026
Let $f\left( x \right) = \left\{ {\matrix{ {x{e^{ax}},\,\,\,\,\,\,\,x \le 0} \cr {x + a{x^2} - {x^3},\,x > 0} \cr } } \right.$

Where a is a positive constant. Find the interval in which $f'(x)$ is increasing.

1996 Q81 JEE Advanced Numerical
14 Mar 2026
Determine the points of maxima and minima of the function
$f\left( x \right) = {1 \over 8}\ell n\,x - bx + {x^2},x > 0,$ where $b \ge 0$ is a constant.
1996 Q82 JEE Advanced Numerical
14 Mar 2026
A curve $y=f(x)$ passes through the point $P(1, 1)$. The normal to the curve at $P$ is $a(y-1)+(x-1)=0$. If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, determine the equation of the curve. Also obtain the area bounded by the $y$-axis, the curve and the normal to the curve at $P$.
1995 Q83 JEE Advanced MCQ
14 Mar 2026
The function $f\left( x \right) = {{in\,\left( {\pi + x} \right)} \over {in\,\left( {e + x} \right)}}$ is
A.
increasing on $\left( {0,\infty } \right)$
B.
decreasing on $\left( {0,\infty } \right)$
C.
increasing on $\left( {0,\pi /e} \right),$ decreasing on $\left( {\pi /e,\infty } \right)$
D.
decreasing on $\left( {0,\pi /e} \right),$ increasing on $\left( {\pi /e,\infty } \right)$
1995 Q84 JEE Advanced MCQ
14 Mar 2026
The slope of the tangent to a curve $y = f\left( x \right)$ at $\left[ {x,\,f\left( x \right)} \right]$ is $2x+1$. If the curve passes through the point $\left( {1,2} \right)$, then the area bounded by the curve, the $x$-axis and the line $x=1$ is
A.
${5 \over 6}$
B.
${6 \over 5}$
C.
${1 \over 6}$
D.
$6$
1995 Q85 JEE Advanced MCQ
14 Mar 2026
On the interval $\left[ {0,1} \right]$ the function ${x^{25}}{\left( {1 - x} \right)^{75}}$ takes its maximum value at the point
A.
$0$
B.
${1 \over 4}$
C.
${1 \over 2}$
D.
${1 \over 3}$
1995 Q86 JEE Advanced Numerical
14 Mar 2026
Let $(h, k)$ be a fixed point, where $h > 0,k > 0.$. A straight line passing through this point cuts the possitive direction of the coordinate axes at the points $P$ and $Q$. Find the minimum area of the triangle $OPQ$, $O$ being the origin.
1994 Q87 JEE Advanced MCQ
14 Mar 2026
Which one of the following curves cut the parabola ${y^2} = 4ax$ at right angles?
A.
${x^2} + {y^2} = {a^2}$
B.
$y = {e^{ - x/2a}}$
C.
$y = ax$
D.
${x^2} = 4ay$
1994 Q88 JEE Advanced MCQ
14 Mar 2026
The function defined by $f\left( x \right) = \left( {x + 2} \right){e^{ - x}}$
A.
decreasing for all $x$
B.
decreasing in $\left( { - \infty , - 1} \right)$ and increasing in $\left( { - 1,\infty } \right)$
C.
increasing for all $x$
D.
decreasing in $\left( { - 1,\infty } \right)$ and increasing in $\left( { - \infty , - 1} \right)$
1994 Q89 JEE Advanced Numerical
14 Mar 2026
The circle ${x^2} + {y^2} = 1$ cuts the $x$-axis at $P$ and $Q$. Another circle with centre at $Q$ and variable radius intersects the first circle at $R$ above the $x$-axis and the line segment $PQ$ at $S$. Find the maximum area of the triangle $QSR$.
1994 Q90 JEE Advanced Numerical
14 Mar 2026
The curve $y = a{x^3} + b{x^2} + cx + 5$, touches the $x$-axis at $P(-2, 0)$ and cuts the $y$ axis at a point $Q$, where its gradient is $3$. Find $a, b, c$.
1994 Q91 JEE Advanced Numerical
14 Mar 2026
Let $P$ be a variable point on the ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$ with foci ${F_1}$ and ${F_2}$. If $A$ is the area of the triangle $P{F_1}{F_2}$ then the maximum value of $A$ is ..........
1994 Q92 JEE Advanced Numerical
14 Mar 2026
Let $C$ be the curve ${y^3} - 3xy + 2 = 0$. If $H$ is the set of points on the curve $C$ where the tangent is horizontal and $V$ is the set of the point on the curve $C$ where the tangent is vertical then $H=$.............. and $V=$ .................
1993 Q93 JEE Advanced MSQ
14 Mar 2026
If $f\left( x \right) = \left\{ {\matrix{ {3{x^2} + 12x - 1,} & { - 1 \le x \le 2} \cr {37 - x} & {2 < x \le 3} \cr } } \right.$ then:
A.
$f(x)$ is increasing on $\left[ { - 1,2} \right]$
B.
$f(x)$ is continues on $\left[ { - 1,3} \right]$
C.
$f'(2)$ does not exist
D.
$f(x)$ has the maximum value at $x=2$
1993 Q94 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 $x=1$.

1993 Q95 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 $x=0$
1992 Q96 JEE Advanced Numerical
14 Mar 2026
What normal to the curve $y = {x^2}$ forms the shortest chord?
1992 Q97 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

1992 Q98 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 $f(x)$.
1991 Q99 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 ?

1990 Q100 JEE Advanced Numerical
14 Mar 2026
Show that $2\sin x + \tan x \ge 3x$ where $0 \le x < {\pi \over 2}$.