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## Application of Derivatives

### Mathematics

#### Previous Years Questions

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Consider the rectangles lying the region $$\left\{ {(x,y) \in R \times R:0\, \le \,x\, \le \,{\pi \over 2}} \right.$$ a...
JEE Advanced 2020 Paper 1 Offline
Which of the following options is the only CORRECT combination?
JEE Advanced 2017 Paper 1 Offline
Which of the following options is the only CORRECT combination?
JEE Advanced 2017 Paper 1 Offline
Which of the following options is the only INCORRECT combination?
JEE Advanced 2017 Paper 1 Offline
The least value of a $$\in R$$ for which $$4a{x^2} + {1 \over x} \ge 1,$$, for all $$x&gt;0$$. is
JEE Advanced 2016 Paper 1 Offline
Let $$f:\left[ {0,1} \right] \to R$$ (the set of all real numbers) be a function. Suppose the function $$f$$ is twice di...
JEE Advanced 2013 Paper 2 Offline
Let $$f:\left[ {0,1} \right] \to R$$ (the set of all real numbers) be a function. Suppose the function $$f$$ is twice di...
JEE Advanced 2013 Paper 2 Offline
Let $$f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}$$ for all $$x \in IR$$ and let $$g\left(... IIT-JEE 2012 Paper 2 Offline Let$$f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}$$for all$$x \in IR$$and let$$g\left(...
IIT-JEE 2012 Paper 2 Offline
Let the function $$g:\left( { - \infty ,\infty } \right) \to \left( { - {\pi \over 2},{\pi \over 2}} \right)$$ be give...
IIT-JEE 2008
The total number of local maxima and local minimum of the function $$f\left( x \right) = \left\{ {\matrix{ {{{\left(... IIT-JEE 2008 Consider the two curves$${C_1}:{y^2} = 4x,\,{C_2}:{x^2} + {y^2} - 6x + 1 = 0$$. Then, IIT-JEE 2008 If a continuous function$$f$$defined on the real line$$R$$, assumes positive and negative values in$$R$$then the eq... IIT-JEE 2007 If a continuous function$$f$$defined on the real line$$R$$, assumes positive and negative values in$$R$$then the eq... IIT-JEE 2007 If a continuous function$$f$$defined on the real line$$R$$, assumes positive and negative values in$$R$$then the eq... IIT-JEE 2007 The tangent to the curve$$y = {e^x}$$drawn at the point$$\left( {c,{e^c}} \right)$$intersects the line joining the ... IIT-JEE 2007 If$$P(x)$$is a polynomial of degree less than or equal to$$2$$and$$S$$is the set of all such polynomials so that ... IIT-JEE 2005 Screening If$$f\left( x \right) = {x^a}\log x$$and$$f\left( 0 \right) = 0,$$then the value of$$\alpha $$for which Rolle's th... IIT-JEE 2004 Screening If$$f\left( x \right) = {x^3} + b{x^2} + cx + d$$and$$0 &lt; {b^2} &lt; c,$$then in$$\left( { - \infty ,\infty } \r...
IIT-JEE 2004 Screening
Tangent is drawn to ellipse $${{{x^2}} \over {27}} + {y^2} = 1\,\,\,at\,\left( {3\sqrt 3 \cos \theta ,\sin \theta } \ri... IIT-JEE 2003 Screening In$$\left[ {0,1} \right]$$Languages Mean Value theorem is NOT applicable to IIT-JEE 2003 Screening The point(s) in the curve$${y^3} + 3{x^2} = 12y$$where the tangent is vertical, is (are) IIT-JEE 2002 Screening The length of a longest interval in which the function$$3\,\sin x - 4{\sin ^3}x$$is increasing, is IIT-JEE 2002 Screening Let$$f\left( x \right) = \left( {1 + {b^2}} \right){x^2} + 2bx + 1$$and let$$m(b)$$be the minimum value of$$f(x)$$... IIT-JEE 2001 Screening The triangle formed by the tangent to the curve$$f\left( x \right) = {x^2} + bx - b$$at the point$$(1, 1)$$and the c... IIT-JEE 2001 Screening If$$f\left( x \right) = x{e^{x\left( {1 - x} \right)}},$$then$$f(x)$$is IIT-JEE 2001 Screening For all$$x \in \left( {0,1} \right)$$IIT-JEE 2000 Screening Let$$f\left( x \right) = \left\{ {\matrix{ {\left| x \right|,} &amp; {for} &amp; {0 &lt; \left| x \right| \le 2} \c...
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 int...
IIT-JEE 2000 Screening
If the normal to the curve $$y = f\left( x \right)$$ and the point $$(3, 4)$$ makes an angle $${{{3\pi } \over 4}}$$ wit...
IIT-JEE 2000 Screening
Consider the following statements in $$S$$ and $$R$$ $$S:$$ $$\,\,\,$$$Both $$\sin \,\,x$$ and $$\cos \,\,x$$ are decr... IIT-JEE 2000 Screening The function $$f(x)=$$ $${\sin ^4}x + {\cos ^4}x$$ increases if IIT-JEE 1999 If $$f\left( x \right) = {{{x^2} - 1} \over {{x^2} + 1}},$$ for every real number $$x$$, then the minimum value of $$f$$... IIT-JEE 1998 The number of values of $$x$$ where the function $$f\left( x \right) = \cos x + \cos \left( {\sqrt 2 x} \right)$$ attai... IIT-JEE 1998 If $$f\left( x \right) = {x \over {\sin x}}$$ and $$g\left( x \right) = {x \over {\tan x}}$$, where $$0 &lt; x \le 1$$, ... IIT-JEE 1997 The slope of the tangent to a curve $$y = f\left( x \right)$$ at $$\left[ {x,\,f\left( x \right)} \right]$$ is $$2x+1$$.... IIT-JEE 1995 Screening On the interval $$\left[ {0,1} \right]$$ the function $${x^{25}}{\left( {1 - x} \right)^{75}}$$ takes its maximum value ... IIT-JEE 1995 Screening The function $$f\left( x \right) = {{in\,\left( {\pi + x} \right)} \over {in\,\left( {e + x} \right)}}$$ is IIT-JEE 1995 Screening Which one of the following curves cut the parabola $${y^2} = 4ax$$ at right angles? IIT-JEE 1994 The function defined by $$f\left( x \right) = \left( {x + 2} \right){e^{ - x}}$$ IIT-JEE 1994 Let $$f$$ and $$g$$ be increasing and decreasing functions, respectively from $$\left[ {0,\infty } \right)$$ to $$\left[... IIT-JEE 1987 The smallest positive root of the equation,$$\tan x - x = 0$$lies in IIT-JEE 1987 Let$$P\left( x \right) = {a_0} + {a_1}{x^2} + {a_2}{x^4} + ...... + {a_n}{x^{2n}}$$be a polynomial in a real variable ... IIT-JEE 1986 If$$y = a\,\,In\,x + b{x^2} + x$$has its extreamum values at$$x=-1$$and$$x=2$$, then IIT-JEE 1983 The normal to the curve$$\,x = a\left( {\cos \theta + \theta \sin \theta } \right)$$,$$y = a\left( {\sin \theta - \t... IIT-JEE 1983 $$AB$$ is a diameter of a circle and $$C$$ is any point on the circumference of the circle. Then IIT-JEE 1983 If $$a+b+c=0$$, then the quadratic equation $$3a{x^2} + 2bx + c = 0$$ has IIT-JEE 1983 ## MCQ (More than One Correct Answer) More Let, $$f(x) = {{\sin \pi x} \over {{x^2}}}$$, x &gt; 0Let x1 &lt; x2 &lt; x3 &lt; ... &lt; xn &lt; ... be all the points... JEE Advanced 2019 Paper 2 Offline Let f : R $$\to$$ R be given by$$f(x) = (x - 1)(x - 2)(x - 5)$$. Define$$F(x) = \int\limits_0^x {f(t)dt}$$, x &gt; 0T... JEE Advanced 2019 Paper 2 Offline If $$f(x) = \left| {\matrix{ {\cos 2x} &amp; {\cos 2x} &amp; {\sin 2x} \cr { - \cos x} &amp; {\cos x} &amp; { - ... JEE Advanced 2017 Paper 2 Offline f : R$$ \to $$R is a differentiable function such that f'(x) &gt; 2f(x) for all x$$ \in $$R, and f(0) = 1 then JEE Advanced 2017 Paper 2 Offline Let f: R$$ \to \left( {0,\infty } \right)$$and g : R$$ \to $$R be twice differentiable functions such that f'' and g... JEE Advanced 2016 Paper 2 Offline Let$$f, g :\left[ { - 1,2} \right] \to R$$be continuous functions which are twice differentiable on the interval ... JEE Advanced 2015 Paper 2 Offline The function$$f(x) = 2\left| x \right| + \left| {x + 2} \right| - \left| {\left| {x + 2} \right| - 2\left| x \right|} \... JEE Advanced 2013 Paper 2 Offline A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $$8:15$$ is converted into an open r... JEE Advanced 2013 Paper 1 Offline If $$f\left( x \right) = \int_0^x {{e^{{t^2}}}} \left( {t - 2} \right)\left( {t - 3} \right)dt$$ for all $$x \in \left( ... IIT-JEE 2012 Paper 2 Offline For the function$$$f\left( x \right) = x\cos \,{1 \over x},x \ge 1,$$IIT-JEE 2009 Let$$f\left( x \right) = \left\{ {\matrix{ {{e^x},} &amp; {0 \le x \le 1} \cr {2 - {e^{x - 1}},} &amp; {1 &lt; ...
IIT-JEE 2006
$$f(x)$$ is cubic polynomial with $$f(2)=18$$ and $$f(1)=-1$$. Also $$f(x)$$ has local maxima at $$x=-1$$ and $$f'(x)$$ ...
IIT-JEE 2006
The function $$f\left( x \right) = \int\limits_{ - 1}^x {t\left( {{e^t} - 1} \right)\left( {t - 1} \right){{\left( {t - ... IIT-JEE 1999 Let$$h\left( x \right) = f\left( x \right) - {\left( {f\left( x \right)} \right)^2} + {\left( {f\left( x \right)} \righ...
IIT-JEE 1998
If $$f\left( x \right) = \left\{ {\matrix{ {3{x^2} + 12x - 1,} &amp; { - 1 \le x \le 2} \cr {37 - x} &amp; {2 &l... IIT-JEE 1993 If the line$$ax+by+c=0$$is a normal to the curve$$xy=1$$, then IIT-JEE 1986 ## Numerical More For each positive integer n, let$${y_n} = {1 \over n}(n + 1)(n + 2)...{(n + n)^{{1 \over n}}}$$. For x$$ \in $$R, let [... JEE Advanced 2018 Paper 1 Offline A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner v... JEE Advanced 2015 Paper 1 Offline The slope of the tangent to the curve$${\left( {y - {x^5}} \right)^2} = x{\left( {1 + {x^2}} \right)^2}$$at the point ... JEE Advanced 2014 Paper 1 Offline Let$$p(x)$$be a real polynomial of least degree which has a local maximum at$$x=1$$and a local minimum at$$x=3$$. I... IIT-JEE 2012 Paper 1 Offline Let$$f:IR \to IR$$be defined as$$f\left( x \right) = \left| x \right| + \left| {{x^2} - 1} \right|.$$The total numbe... IIT-JEE 2012 Paper 1 Offline Let$$f$$be a real-valued differentiable function on$$R$$(the set of all real numbers) such that$$f(1)=1$$. If the ... IIT-JEE 2010 Paper 1 Offline Let$$f$$be a function defined on$$R$$(the set of all real numbers) such that$$f'\left( x \right) = 2010\left( {x -...
IIT-JEE 2010 Paper 2 Offline
The maximum value of the function $$f\left( x \right) = 2{x^3} - 15{x^2} + 36x - 48$$ on the set $${\rm A} = \left\{ {x... IIT-JEE 2009 The maximum value of the function$$f\left( x \right) = 2{x^3} - 15{x^2} + 36x - 48$$on the set$${\rm A} = \left\{ {x...
IIT-JEE 2009
Let $$p(x)$$ be a polynomial of degree $$4$$ having extremum at $$x = 1,2$$ and $$\mathop {\lim }\limits_{x \to 0} \lef... IIT-JEE 2009 ## Subjective More For a twice differentiable function$$f(x),g(x)$$is defined as$$4\sqrt {65} g\left( x \right) = \left( {f'{{\left( x \...
IIT-JEE 2006
If $$p(x)$$ be a polynomial of degree $$3$$ satisfying $$p(-1)=10, p(1)=-6$$ and $$p(x)$$ has maxima at $$x=-1$$ and $$p... IIT-JEE 2005 If$$\left| {f\left( {{x_1}} \right) - f\left( {{x_2}} \right)} \right| &lt; {\left( {{x_1} - {x_2}} \right)^2},$$for ... IIT-JEE 2005 Prove that for$$x \in \left[ {0,{\pi \over 2}} \right],\sin x + 2x \ge {{3x\left( {x + 1} \right)} \over \pi }$$.... IIT-JEE 2004 Using Rolle's theorem, prove that there is at least one root in$$\left( {{{45}^{1/100}},46} \right)$$of the polynomia... IIT-JEE 2004 If$$P(1)=0$$and$${{dp\left( x \right)} \over {dx}} &gt; P\left( x \right)$$for all$$x \ge 1$$then prove that$$P(...
IIT-JEE 2003
If the function $$f:\left[ {0,4} \right] \to R$$ is differentiable then show that (i)$$\,\,\,\,\,$$ For $$a, b$$$$\,\,... IIT-JEE 2003 Find a point on the curve$${x^2} + 2{y^2} = 6$$whose distance from the line$$x+y=7$$, is minimum. IIT-JEE 2003 Using the relation$$2\left( {1 - \cos x} \right) &lt; {x^2},\,x \ne 0$$or otherwise, prove that$$\sin \left( {\tan x...
IIT-JEE 2003
Let $$- 1 \le p \le 1$$. Show that the equation $$4{x^3} - 3x - p = 0$$ has a unique root in the interval $$\left[ {1/... IIT-JEE 2001 Suppose$$p\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + .......... + {a_n}{x^n}.$$If$$\left| {p\left( x \right)} ...
IIT-JEE 2000
Suppose $$f(x)$$ is a function satisfying the following conditions (a) $$f(0)=2,f(1)=1$$, (b) $$f$$has a minimum value ...
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 $$A$$...
IIT-JEE 1998
Let $$a+b=4$$, where $$a&lt;2,$$ and let $$g(x)$$ be a differentiable function. If $${{dg} \over {dx}} &gt; 0$$ for all ...
IIT-JEE 1997
Determine the points of maxima and minima of the function $$f\left( x \right) = {1 \over 8}\ell n\,x - bx + {x^2},x &gt... IIT-JEE 1996 A curve$$y=f(x)$$passes through the point$$P(1, 1)$$. The normal to the curve at$$P$$is$$a(y-1)+(x-1)=0$$. If the ... IIT-JEE 1996 Let$$f\left( x \right) = \left\{ {\matrix{ {x{e^{ax}},\,\,\,\,\,\,\,x \le 0} \cr {x + a{x^2} - {x^3},\,x &gt; 0...
IIT-JEE 1996
Let $$(h, k)$$ be a fixed point, where $$h &gt; 0,k &gt; 0.$$. A straight line passing through this point cuts the possi...
IIT-JEE 1995
The circle $${x^2} + {y^2} = 1$$ cuts the $$x$$-axis at $$P$$ and $$Q$$. Another circle with centre at $$Q$$ and variabl...
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 $$... IIT-JEE 1994 Let$$f\left( x \right) = \left\{ {\matrix{ { - {x^3} + {{\left( {{b^3} - {b^2} + b - 1} \right)} \over {\left( {{b^2...
IIT-JEE 1993
Find the equation of the normal to the curve $$y = {\left( {1 + x} \right)^y} + {\sin ^{ - 1}}\left( {{{\sin }^2}x} \ri... IIT-JEE 1993 In this questions there are entries in columns$$I$$and$$II$$. Each entry in column$$I$$is related to exactly one en... IIT-JEE 1992 What normal to the curve$$y = {x^2}$$forms the shortest chord? IIT-JEE 1992 A cubic$$f(x)$$vanishes at$$x=2$$and has relative minimum / maximum at$$x=-1$$and$$x = {1 \over 3}$$if$$\int\li...
IIT-JEE 1992
A window of perimeter $$P$$ (including the base of the arch) is in the form of a rectangle surmounded by a semi circle. ...
IIT-JEE 1991
A point $$P$$ is given on the circumference of a circle of radius $$r$$. Chord $$QR$$ is parallel to the tangent at $$P... IIT-JEE 1990 Show that$$2\sin x + \tan x \ge 3x$$where$$0 \le x &lt; {\pi \over 2}$$. IIT-JEE 1990 Find all maxima and minima of the function$$$y = x{\left( {x - 1} \right)^2},0 \le x \le 2$$Also determine the area ... IIT-JEE 1989 Investigate for maxima and minimum the function$$$f\left( x \right) = \int\limits_1^x {\left[ {2\left( {t - 1} \right...
IIT-JEE 1988
Find the point on the curve $$\,\,\,4{x^2} + {a^2}{y^2} = 4{a^2},\,\,\,4 &lt; {a^2} &lt; 8$$ that is farthest from the ...
IIT-JEE 1987
Let $$f\left( x \right) = {\sin ^3}x + \lambda {\sin ^2}x, - {\pi \over 2} &lt; x &lt; {\pi \over 2}.$$ Find the inter...
IIT-JEE 1985
Find all the tangents to the curve $$y = \cos \left( {x + y} \right),\,\, - 2\pi \le x \le 2\pi ,$$ that are parallel ...
IIT-JEE 1985
Find the coordinates of the point on the curve $$y = {x \over {1 + {x^2}}}$$ where the tangent to the curve has the gre...
IIT-JEE 1983
Show that $$1+x$$ $$In\left( {x + \sqrt {{x^2} + 1} } \right) \ge \sqrt {1 + {x^2}}$$ for all $$x \ge 0$$
IIT-JEE 1983
If $$a{x^2} + {b \over x} \ge c$$ for all positive $$x$$ where $$a&gt;0$$ and $$b&gt;0$$ show that $$27a{b^2} \ge 4{c^3}... IIT-JEE 1982 If$$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$$;... IIT-JEE 1982 For all$$x$$in$$\left[ {0,1} \right]$$, let the second derivative$$f''(x)$$of a function$$f(x)$$exist and satisfy... IIT-JEE 1981 Let$$x$$and$$y$$be two real variables such that$$x&gt;0$$and$$xy=1$$. Find the minimum value of$$x+y$$. IIT-JEE 1981 Use the function$$f\left( x \right) = {x^{1/x}},x &gt; 0$$. to determine the bigger of the two numbers$${e^\pi }$$and... IIT-JEE 1981 Prove that the minimum value of$${{\left( {a + x} \right)\left( {b + x} \right)} \over {\left( {c + x} \right)}},a...
IIT-JEE 1979

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Let $$P$$ be a variable point on the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ with foci $${F_1}... IIT-JEE 1994 Let$$C$$be the curve$${y^3} - 3xy + 2 = 0$$. If$$H$$is the set of points on the curve$$C$$where the tangent is ho... IIT-JEE 1994 The set of all$$x$$for which$$in\left( {1 + x} \right) \le x$$is equal to .......... IIT-JEE 1987 The function$$y = 2{x^2} - In\,\left| x \right|$$is monotonically increasing for values of$$x\left( {x \ne 0} \right)...
IIT-JEE 1983
The larger of $$\cos \left( {In\,\,\theta } \right)$$ and $$In$$ $$\left( {\cos \,\,\theta } \right)$$ If $${e^{ - \pi... IIT-JEE 1983 ## True or False More For$$0 &lt; a &lt; x,$$the minimum value of the function$$lo{g_a}x + {\log _x}a$$is$$2$$. IIT-JEE 1984 If$$x-r$$is a factor of the polynomial$$f\left( x \right) = {a_n}{x^4} + ..... + {a_0},$$repeated$$m$$times$$\lef...
IIT-JEE 1983

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