Application of Derivatives
127 Questions
2004
JEE Advanced
Numerical
IIT-JEE 2004
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.
the identity if any used in the proof.
Correct Answer: Solve it.
2004
JEE Advanced
Numerical
IIT-JEE 2004
Using Rolle's theorem, prove that there is at least one root
in $\left( {{{45}^{1/100}},46} \right)$ of the polynomial
$P\left( x \right) = 51{x^{101}} - 2323{\left( x \right)^{100}} - 45x + 1035$.
in $\left( {{{45}^{1/100}},46} \right)$ of the polynomial
$P\left( x \right) = 51{x^{101}} - 2323{\left( x \right)^{100}} - 45x + 1035$.
Correct Answer: Solve it.
2003
JEE Advanced
MCQ
IIT-JEE 2003 Screening
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
JEE Advanced
MCQ
IIT-JEE 2003 Screening
Tangent is drawn to ellipse
${{{x^2}} \over {27}} + {y^2} = 1\,\,\,at\,\left( {3\sqrt 3 \cos \theta ,\sin \theta } \right)\left( {where\,\,\theta \in \left( {0,\pi /2} \right)} \right)$.
${{{x^2}} \over {27}} + {y^2} = 1\,\,\,at\,\left( {3\sqrt 3 \cos \theta ,\sin \theta } \right)\left( {where\,\,\theta \in \left( {0,\pi /2} \right)} \right)$.
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
JEE Advanced
Numerical
IIT-JEE 2003
Find a point on the curve ${x^2} + 2{y^2} = 6$ whose distance from
the line $x+y=7$, is minimum.
the line $x+y=7$, is minimum.
Correct Answer: $$(2, 1)$$
2003
JEE Advanced
Numerical
IIT-JEE 2003
Using the relation $2\left( {1 - \cos x} \right) < {x^2},\,x \ne 0$ or otherwise,
prove that $\sin \left( {\tan x} \right) \ge x,\,\forall x \in \left[ {0,{\pi \over 4}} \right]$
prove that $\sin \left( {\tan x} \right) \ge x,\,\forall x \in \left[ {0,{\pi \over 4}} \right]$
Correct Answer: Solve it.
2003
JEE Advanced
Numerical
IIT-JEE 2003
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} $
(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} $
Correct Answer: Solve it.
2003
JEE Advanced
Numerical
IIT-JEE 2003
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 $x>1$.
$P(x)>0$ for all $x>1$.
Correct Answer: Solve it.
2002
JEE Advanced
MCQ
IIT-JEE 2002 Screening
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
JEE Advanced
MCQ
IIT-JEE 2002 Screening
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
JEE Advanced
MCQ
IIT-JEE 2001 Screening
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
JEE Advanced
MCQ
IIT-JEE 2001 Screening
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
JEE Advanced
MCQ
IIT-JEE 2001 Screening
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
JEE Advanced
Numerical
IIT-JEE 2001
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.
has a unique root in the interval $\left[ {1/2,\,1} \right]$ and identify it.
Correct Answer: Solve it.
2000
JEE Advanced
MCQ
IIT-JEE 2000 Screening
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 ?
$S:$ $\,\,\,$$ Both $\sin \,\,x$ and $\cos \,\,x$ are decreasing functions in the interval $\left( {{\pi \over 2},\pi } \right)$
$R:$$\,\,\,$ 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
JEE Advanced
MCQ
IIT-JEE 2000 Screening
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
JEE Advanced
MCQ
IIT-JEE 2000 Screening
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
JEE Advanced
MCQ
IIT-JEE 2000 Screening
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
JEE Advanced
MCQ
IIT-JEE 2000 Screening
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
JEE Advanced
Numerical
IIT-JEE 2000
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
$\left| {{a_1} + 2{a_2} + ........ + n{a_n}} \right| \le 1$.
$\left| {p\left( x \right)} \right| \le \left| {{e^{x - 1}} - 1} \right|$ for all $x \ge 0$, prove that
$\left| {{a_1} + 2{a_2} + ........ + n{a_n}} \right| \le 1$.
Correct Answer: Solve it.
1999
JEE Advanced
MCQ
IIT-JEE 1999
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
JEE Advanced
MSQ
IIT-JEE 1999
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
JEE Advanced
MCQ
IIT-JEE 1998
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
JEE Advanced
MCQ
IIT-JEE 1998
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
$f\left( x \right) = \cos x + \cos \left( {\sqrt 2 x} \right)$ attains its maximum is
A.
$0$
B.
$1$
C.
$2$
D.
infinite
1998
JEE Advanced
MSQ
IIT-JEE 1998
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
JEE Advanced
Numerical
IIT-JEE 1998
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 $f(x)$.
(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 $f(x)$.
Correct Answer: $$a = {1 \over 4},\,\,b = - {5 \over 4}$$
<br>$$f\left( x \right) = {1 \over 4}{x^2} - {5 \over 4}x + 2$$
1998
JEE Advanced
Numerical
IIT-JEE 1998
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.
Correct Answer: $$xy=1$$
1997
JEE Advanced
MCQ
IIT-JEE 1997
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
JEE Advanced
Numerical
IIT-JEE 1997
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.
Correct Answer: Solve it.
1996
JEE Advanced
Numerical
IIT-JEE 1996
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.
Correct Answer: $$\left( { - {2 \over a},{a \over 3}} \right)$$
1996
JEE Advanced
Numerical
IIT-JEE 1996
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.
$f\left( x \right) = {1 \over 8}\ell n\,x - bx + {x^2},x > 0,$ where $b \ge 0$ is a constant.
Correct Answer: min at $$x = {1 \over 4}\left( {b + \sqrt {{b^2} - 1} } \right)$$
<br>max at $$x = {1 \over 4}\left( {b - \sqrt {{b^2} - 1} } \right)$$
1996
JEE Advanced
Numerical
IIT-JEE 1996
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 $P$.
Correct Answer: $$y = {e^{a\left( {x - 1} \right)}}$$
<br> Area $$=$$ $$1$$ sq. unit.
1995
JEE Advanced
MCQ
IIT-JEE 1995 Screening
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
JEE Advanced
MCQ
IIT-JEE 1995 Screening
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
JEE Advanced
MCQ
IIT-JEE 1995 Screening
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
JEE Advanced
Numerical
IIT-JEE 1995
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.
Correct Answer: $$2$$ $$kh$$
1994
JEE Advanced
MCQ
IIT-JEE 1994
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
JEE Advanced
MCQ
IIT-JEE 1994
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
JEE Advanced
Numerical
IIT-JEE 1994
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 $QSR$.
Correct Answer: $${{4\sqrt 3 } \over 9}$$ sq. units
1994
JEE Advanced
Numerical
IIT-JEE 1994
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 $a, b, c$.
Correct Answer: $$a = - {1 \over 2},b = - {3 \over 4},c = 3$$
1994
JEE Advanced
Numerical
IIT-JEE 1994
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 ..........
Correct Answer: $$abc$$
1994
JEE Advanced
Numerical
IIT-JEE 1994
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 $V=$ .................
Correct Answer: $$H = \phi $$
<br>$$V = \left\{ {\left( {1,1} \right)} \right\}$$
1993
JEE Advanced
MSQ
IIT-JEE 1993
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
JEE Advanced
Numerical
IIT-JEE 1993
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
JEE Advanced
Numerical
IIT-JEE 1993
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
JEE Advanced
Numerical
IIT-JEE 1992
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
JEE Advanced
Numerical
IIT-JEE 1992
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
JEE Advanced
Numerical
IIT-JEE 1992
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
JEE Advanced
Numerical
IIT-JEE 1991
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
JEE Advanced
Numerical
IIT-JEE 1990
Show that $2\sin x + \tan x \ge 3x$ where $0 \le x < {\pi \over 2}$.
Correct Answer: Solve it.