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Application of Derivatives

570 Questions
All Exams JEE Mains JEE Advanced TS-EAMCET AP-EAPCET
2002 JEE Mains MCQ
AIEEE 2002
If $2a+3b+6c=0,$ $\left( {a,b,c \in R} \right)$ then the quadratic equation $a{x^2} + bx + c = 0$ has
A.
at least one root in $\left[ {0,1} \right]$
B.
at least one root in $\left[ {2,3} \right]$
C.
at least one root in $\left[ {4,5} \right]$
D.
none of these
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.
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 ?
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
$\left| {{a_1} + 2{a_2} + ........ + n{a_n}} \right| \le 1$.
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
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 $f(x)$.
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.
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.

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.

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.
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 $P$.
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.
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 $QSR$.
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 $a, b, c$.
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 ..........
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 $V=$ .................
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 $x=1$.

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 $x=0$
1992 JEE Advanced Numerical
IIT-JEE 1992
What normal to the curve $y = {x^2}$ forms the shortest chord?
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

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 $f(x)$.
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 ?

1990 JEE Advanced Numerical
IIT-JEE 1990
Show that $2\sin x + \tan x \ge 3x$ where $0 \le x < {\pi \over 2}$.
1990 JEE Advanced Numerical
IIT-JEE 1990
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 $PQR$.
1989 JEE Advanced Numerical
IIT-JEE 1989
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 $y-2$.
1988 JEE Advanced Numerical
IIT-JEE 1988
Investigate for maxima and minimum the function $$f\left( x \right) = \int\limits_1^x {\left[ {2\left( {t - 1} \right){{\left( {t - 2} \right)}^3} + 3{{\left( {t - 1} \right)}^2}{{\left( {t - 2} \right)}^2}} \right]} dt$$
1987 JEE Advanced MCQ
IIT-JEE 1987
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 JEE Advanced MCQ
IIT-JEE 1987
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 JEE Advanced Numerical
IIT-JEE 1987
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 $(0, -2)$.
1987 JEE Advanced Numerical
IIT-JEE 1987
The set of all $x$ for which $in\left( {1 + x} \right) \le x$ is equal to ..........