Maxima and Minima

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JEE Main 2016

LEVELJEE Main

A wire of length $2$ units is cut into two parts which are bent respectively to form a square of side $= x$ units and a circle of radius $= r$ units. If the sum of the areas of the square and the circle so formed is minimum, then

Let $f (x)$ be a polynomial of degree four having extreme values at $x = 1$ and $x = 2$ . If $lim_{x \to 0} [1 + \frac{f ( x )}{x ^{2}}] = 3$ , then $f (2)$ is equal to

If $x = - 1$ and $x = 2$ are extreme points of $f (x) = α lo g ∣ x ∣ + β x^{2} + x$ then

(A)

α = 2, β = - \frac{1}{2}

(B)

α = 2, β = \frac{1}{2}

(C)

α = - 6, β = \frac{1}{2}

(D)

α = - 6, β = - \frac{1}{2}

JEE Main 2012

LEVELJEE Main

Let $a, b \in R$ be such that the function $f$ given by $f (x) = ln ∣ x ∣ + b x^{2} + a x, x \neq = 0$ has extreme values at $x = - 1$ and $x = 2$ . \\ \textbf{Statement-1:} $f$ has local maximum at $x = - 1$ and at $x = 2$ . \\ \textbf{Statement-2:} $a = 1/2$ and $b = - 1/4$ .

(A)

Statement-1 is false, Statement-2 is true.

(B)

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

(C)

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(D)

Statement-1 is true, Statement-2 is false.

JEE Main 2010

LEVELJEE Main

Let $f : R \to R$ be defined by $f (x) = {k - 2 x, 2 x + 3, if x \leq - 1 if x > - 1$ . If $f$ has a local minimum at $x = - 1$ , then a possible value of $k$ is

Let $f : R \to R$ be a continuous function defined by $f (x) = \frac{1}{e ^{x} + 2 e ^{- x}}$ . \\ \textbf{Statement-1:} $f (c) = 1/3$ for some $c \in R$ . \\ \textbf{Statement-2:} $0 < f (x) \leq 1/ (22)$ for all $x \in R$ .

(A)

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(B)

Statement-1 is true, Statement-2 is false.

(C)

Statement-1 is false, Statement-2 is true.

(D)

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

JEE Main 2009

LEVELJEE Main

Given $P (x) = x^{4} + a x^{3} + b x^{2} + c x + d$ such that $x = 0$ is the only real root of $P^{'} (x) = 0$ . If $P (- 1) < P (1)$ , then in the interval $[- 1, 1]$ :

(A)

P (- 1)

is not minimum but

P (1)

is the maximum of

P

(B)

P (- 1)

is the minimum but

P (1)

is not the maximum of

P

(C)

Neither

P (- 1)

is the minimum nor

P (1)

is the maximum of

P

(D)

P (- 1)

is the minimum and

P (1)

is the maximum of

P

JEE Main 2008

LEVELJEE Main

Suppose the cubic $x^{3} - p x + q$ has three distinct real roots where $p > 0$ and $q > 0$ . Then which one of the following holds?

(A)

The cubic has minima at

p /3

and maxima at

- p /3

(B)

The cubic has minima at

- p /3

and maxima at

p /3

(C)

The cubic has minima at both

p /3

and

- p /3

(D)

The cubic has maxima at both

p /3

and

- p /3

JEE Main 2006

LEVELJEE Main

The function $f (x) = \frac{x}{2} + \frac{2}{x}$ has a local minimum at

(A)

x = 2

(B)

x = - 2

(C)

x = 0

(D)

x = 1

JEE Main 2006

LEVELJEE Main

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $x$ . The maximum area enclosed by the park is

(A)

\frac{3}{2} x^{2}

(B)

\frac{x ^{3}}{8}

(C)

\frac{1}{2} x^{2}

(D)

π x^{2}

JEE Main 2005

LEVELJEE Main

Area of the greatest rectangle that can be inscribed in the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ is

ab

The real number $x$ when added to its inverse gives the minimum value of the sum at $x$ equal to

(A)

- 2

(B)

(C)

(D)

- 1

JEE Main 2003

LEVELJEE Main

If the function $f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1$ , where $a > 0$ , attains its maximum and minimum at $p$ and $q$ respectively such that $p^{2} = q$ , then $a$ equals

Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity $u$ and the other from rest with uniform acceleration $f$ . Let $α$ be the angle between their directions of motion. The relative velocity of the second particle w.r.t. the first is least after a time

(A)

\frac{u c o s α}{f}

(B)

\frac{u s i n α}{f}

(C)

\frac{f c o s α}{u}

(D)

u sin α

JEE Main 2002

LEVELJEE Main

The maximum distance from origin of a point on the curve $x = a sin t - b sin (\frac{a t}{b})$ , $y = a cos t - b cos (\frac{a t}{b})$ , both $a, b > 0$ is

a^{2} + b^{2}

(D)

a^{2} - b^{2}

JEE Advanced 2016

LEVELJEE Main

The least value of $a \in R$ for which $4 α x^{2} + \frac{1}{x} \geq 1$ , for all $x > 0$ , is

(A)

\frac{1}{64}

(B)

\frac{1}{32}

(C)

\frac{1}{27}

(D)

\frac{1}{25}

JEE Advanced 2016

LEVELJEE Main

Let $f : R \to (0, \infty)$ and $g : R \to R$ be twice differentiable functions such that $f^{''}$ and $g^{''}$ are continuous functions on $R$ . Suppose $f^{'} (2) = g (2) = 0, f^{''} (2) \neq = 0$ and $g^{'} (2) \neq = 0$ . If $lim_{x \to 2} \frac{f ( x ) g ( x )}{f ^{'} ( x ) g ^{'} ( x )} = 1$ , then

* Multiple Correct Options

(A)

f

has a local minimum at

x = 2

(B)

f

has a local maximum at

x = 2

(C)

f^{''} (2) > f (2)

(D)

f (x) - f^{''} (x) = 0

for at least one

x \in R

JEE Advanced 2014

LEVELJEE Advanced

Let $a \in R$ and let $f : R \to R$ be given by $f (x) = x^{5} - 5 x + a$ . Then

* Multiple Correct Options

(A)

f (x)

has three real roots if

a > 4

(B)

f (x)

has only real root if

a > 4

(C)

f (x)

has three real roots if

a < - 4

(D)

f (x)

has three real roots if

- 4 < a < 4

JEE Advanced 2013

LEVELJEE Advanced

The function $f (x) = 2∣ x ∣ + ∣ x + 2∣ - ∣∣ x + 2∣ - 2∣ x ∣∣$ has a local minimum or a local maximum at $x =$

* Multiple Correct Options

(A)

- 2

(B)

- 2/3

(C)

2

(D)

2/3

JEE Advanced 2013

LEVELJEE Main

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8 : 15$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is $100$ , the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are

* Multiple Correct Options

If $f (x) = \int_{0}^{x^{2}} e^{t} (t - 2) (t - 3) d t$ for all $x \in (0, \infty)$ , then

* Multiple Correct Options

(A)

f

has a local maximum at

x = 2

(B)

f

is decreasing on

(2, 3)

(C)

there exists some

c \in (0, \infty)

, such that

f^{''} (c) = 0

(D)

f

has a local minimum at

x = 3

JEE Advanced 2010

LEVELJEE Main

Let $f, g$ and $h$ be real-valued functions defined on the interval $[0, 1]$ by $f (x) = e^{x^{2}} + e^{- x^{2}}, g (x) = x e^{x^{2}} + e^{- x^{2}}$ and $h (x) = x^{2} e^{x^{2}} + e^{- x^{2}}$ . If $a, b$ and $c$ denote, respectively, the absolute maximum of $f, g$ and $h$ on $[0, 1]$ , then

(A)

a = b

and

c \neq = b

(B)

a = c

and

a \neq = b

(C)

a \neq = b

and

c \neq = b

(D)

a = b = c

JEE Advanced 2008

LEVELJEE Main

The total number of local maxima and local minima of the function $f (x) = {(2 + x)^{3}, x^{2/3}, - 3 < x \leq - 1 - 1 < x < 2$ is

(A)

0

(B)

1

(C)

2

(D)

3

JEE Advanced 2006

LEVELJEE Main

$f (x)$ is cubic polynomial with $f (2) = 18$ and $f (1) = - 1$ . Also $f (x)$ has local maxima at $x = - 1$ and $f^{'} (x)$ has local minima at $x = 0$ , then

* Multiple Correct Options

(A)

the distance between

(- 1, 2)

and

(a, f (a))

, where

x = a

is the point of local minima is

25

(B)

f (x)

is increasing for

x \in [1, 25]

(C)

f (x)

has local minima at

x = 1

(D)

the value of

f (0) = 15

JEE Advanced 2006

LEVELJEE Advanced

Let $f (x) = ⎩ ⎨ ⎧ e^{x}, 2 - e^{x - 1}, x - e, 0 \leq x \leq 1 1 < x \leq 2 2 < x \leq 3$ and $g (x) = \int_{0}^{x} f (t) d t, x \in [1, 3]$ then $g (x)$ has

(A)

local maxima at

x = 1 + ln 2

and local minima at

x = e

(B)

local maxima at

x = 1

and local minima at

x = 2

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x) be a polynomial of degree four having extreme values at x=1 and x=2. If limx→0​[1+x2f(x)​]=3, then f(2) is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x=−1 and x=2 are extreme points of f(x)=αlog∣x∣+βx2+x then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be defined by f(x)={k−2x,2x+3,​if x≤−1if x>−1​. If f has a local minimum at x=−1, then a possible value of k is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Given P(x)=x4+ax3+bx2+cx+d such that x=0 is the only real root of P′(x)=0. If P(−1)<P(1), then in the interval [−1,1]:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Suppose the cubic x3−px+q has three distinct real roots where p>0 and q>0. Then which one of the following holds?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The function f(x)=2x​+x2​ has a local minimum at

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Area of the greatest rectangle that can be inscribed in the ellipse a2x2​+b2y2​=1 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The real number x when added to its inverse gives the minimum value of the sum at x equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the function f(x)=2x3−9ax2+12a2x+1, where a>0, attains its maximum and minimum at p and q respectively such that p2=q, then a equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The maximum distance from origin of a point on the curve x=asint−bsin(bat​), y=acost−bcos(bat​), both a,b>0 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The least value of a∈R for which 4αx2+x1​≥1, for all x>0, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a∈R and let f:R→R be given by f(x)=x5−5x+a. Then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The function f(x)=2∣x∣+∣x+2∣−∣∣x+2∣−2∣x∣∣ has a local minimum or a local maximum at x=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)=∫0x2​et​(t−2)(t−3)dt for all x∈(0,∞), then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The total number of local maxima and local minima of the function f(x)={(2+x)3,x2/3,​−3<x≤−1−1<x<2​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 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) has local minima at x=0, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)=⎩⎨⎧​ex,2−ex−1,x−e,​0≤x≤11<x≤22<x≤3​ and g(x)=∫0x​f(t)dt,x∈[1,3] then g(x) has

Let $f (x)$ be a polynomial of degree four having extreme values at $x = 1$ and $x = 2$ . If $lim_{x \to 0} [1 + \frac{f ( x )}{x ^{2}}] = 3$ , then $f (2)$ is equal to

If $x = - 1$ and $x = 2$ are extreme points of $f (x) = α lo g ∣ x ∣ + β x^{2} + x$ then

Let $f : R \to R$ be defined by $f (x) = {k - 2 x, 2 x + 3, if x \leq - 1 if x > - 1$ . If $f$ has a local minimum at $x = - 1$ , then a possible value of $k$ is

Given $P (x) = x^{4} + a x^{3} + b x^{2} + c x + d$ such that $x = 0$ is the only real root of $P^{'} (x) = 0$ . If $P (- 1) < P (1)$ , then in the interval $[- 1, 1]$ :

Suppose the cubic $x^{3} - p x + q$ has three distinct real roots where $p > 0$ and $q > 0$ . Then which one of the following holds?

The function $f (x) = \frac{x}{2} + \frac{2}{x}$ has a local minimum at

Area of the greatest rectangle that can be inscribed in the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ is

The real number $x$ when added to its inverse gives the minimum value of the sum at $x$ equal to

If the function $f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1$ , where $a > 0$ , attains its maximum and minimum at $p$ and $q$ respectively such that $p^{2} = q$ , then $a$ equals

The maximum distance from origin of a point on the curve $x = a sin t - b sin (\frac{a t}{b})$ , $y = a cos t - b cos (\frac{a t}{b})$ , both $a, b > 0$ is

The least value of $a \in R$ for which $4 α x^{2} + \frac{1}{x} \geq 1$ , for all $x > 0$ , is

Let $a \in R$ and let $f : R \to R$ be given by $f (x) = x^{5} - 5 x + a$ . Then

The function $f (x) = 2∣ x ∣ + ∣ x + 2∣ - ∣∣ x + 2∣ - 2∣ x ∣∣$ has a local minimum or a local maximum at $x =$

If $f (x) = \int_{0}^{x^{2}} e^{t} (t - 2) (t - 3) d t$ for all $x \in (0, \infty)$ , then

The total number of local maxima and local minima of the function $f (x) = {(2 + x)^{3}, x^{2/3}, - 3 < x \leq - 1 - 1 < x < 2$ is

$f (x)$ is cubic polynomial with $f (2) = 18$ and $f (1) = - 1$ . Also $f (x)$ has local maxima at $x = - 1$ and $f^{'} (x)$ has local minima at $x = 0$ , then

Let $f (x) = ⎩ ⎨ ⎧ e^{x}, 2 - e^{x - 1}, x - e, 0 \leq x \leq 1 1 < x \leq 2 2 < x \leq 3$ and $g (x) = \int_{0}^{x} f (t) d t, x \in [1, 3]$ then $g (x)$ has