Mathematics

Maxima and Minima

191 Questions

Maxima and minima problems involve finding the highest and lowest values of mathematical functions within given intervals. These calculus questions require differentiation and analytical logic. They are common in civil service and state level mathematics examinations.

Extreme function valuesComplex number minimizationMinimum variables calculationMaximum matrix analysisCalculus optimization

Maxima and Minima Questions

Multiple choice maths solving equations numerically finding roots by iteration fundamental theorem of algebra complex numbers and linear inequations

If $f\left( {x,y} \right) = \sqrt {{x^2} + {y^2}}  + \sqrt {{{\left( {x - 1} \right)}^2} + {y^2}}  + \sqrt {{x^2} + {{\left( {y - 1} \right)}^2}}  + \sqrt {{{\left( {x - 3} \right)}^2} + {{\left( {y - 4} \right)}^2}} $ where $x,y \in R$, then the minimum value of $f\left( {x,y} \right)$ is

  1. $2 + \sqrt 5 $
  2. $5 + \sqrt 2 $
  3. $5 - \sqrt 2 $
  4. $\sqrt 5 - 2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The function represents the sum of distances from (x, y) to four points: (0, 0), (1, 0), (0, 1), and (3, 4). This is a Fermat point problem. The minimum distance sum for these points is found by connecting the diagonals, resulting in 2 + sqrt(5).

Multiple choice maths equivalent fractions comparing and ordering fractions comparing fractions fractions and its related operations

Which is the greatest number in the following

  1. $16\frac{2}{3}\% $
  2. $\frac{2}{{15}}$
  3. $\frac{1}{{11}}$
  4. $0.17$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
$16 \dfrac 2 3 \% ; \dfrac{2}{15} ; \dfrac{1}{11} ; 0.17$

$\Rightarrow \dfrac{50}{3} \times \dfrac{1}{100} ; \dfrac{2}{15} ; 0.0909.... ; 0.17$

$\Rightarrow \dfrac{1}{3 \times 2} ; 0.1333 ; 0.0909 .... ; 0.17$

$\Rightarrow 0.1666...7 ; 0.1333 ; 0.0909 ; 0.17$

$\therefore 0.17$ is greater 
Multiple choice maths theory of equations forming quadratic equation vieta’s formula for quadratic equations properties of roots of a quadratic equations

If $P ( \alpha , \beta )$ moves on $x ^ { 2 } + y ^ { 2 } - 2 x + 6 y + 1 = 0$ then minimum value of $a ^ { 2 } + \beta ^ { 2 } - 2 a - 4 \beta$ is 

  1. -3

  2. -1

  3. 1

  4. 3
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The circle is x^2 - 2x + 1 + y^2 + 6y + 9 = -1 + 1 + 9, which is (x-1)^2 + (y+3)^2 = 9. The center is (1, -3) and radius is 3. We want the minimum of a^2 + b^2 - 2a - 4b. This is (a-1)^2 + (b-2)^2 - 5. The point (a, b) is on the circle. The minimum distance from (1, 2) to the circle (center (1, -3), radius 3) is |distance between (1, 2) and (1, -3)| - radius = |2 - (-3)| - 3 = 5 - 3 = 2. The minimum value of the squared distance (a-1)^2 + (b-2)^2 is 2^2 = 4. Thus, 4 - 5 = -1.

Multiple choice finding nth roots of a complex number n th root of unity demoivre's theorem complex numbers maths

Suppose A is a complex number and $ n \in N, $ such that $A^{n} = (A + 1)^{n} =1, $ then the least value of $n$ is

  1. $3$
  2. $6$
  3. $9$
  4. $12$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$Since\quad { z }^{ n }=1\ \Rightarrow \quad { \left| z \right|  }^{ n }=1\ \quad \quad \quad \left| z \right| =1\ similarly,\ { \left( z+1 \right)  }^{ n }=1\ \Rightarrow \quad { \left| z+1 \right|  }^{ n }=1\ \left| z+1 \right| =1\ Let\quad z=a+ib\ \left| z \right| =\left| z+1 \right| \quad \Rightarrow \quad { a }^{ 2 }+{ b }^{ 2 }={ \left( a+1 \right)  }^{ 2 }+{ b }^{ 2 }\ { a }^{ 2 }+{ b }^{ 2 }={ a }^{ 2 }+{ b }^{ 2 }+2a+1\ \Rightarrow \quad 2a+1=0\ \therefore \quad a=\frac { -1 }{ 2 } \ putting\quad the\quad value\quad of\quad a\quad in\quad eq.\ \Rightarrow \quad { \left( \frac { -1 }{ 2 }  \right)  }^{ 2 }+{ b }^{ 2 }=1\ \Rightarrow \quad { b }^{ 2 }=\frac { 3 }{ 4 } \ \Rightarrow \quad b=\pm \frac { \sqrt { 3 }  }{ 2 } \ Now,\quad z+1=\quad \frac { 1 }{ 2 } \pm \frac { \sqrt { 3 }  }{ 2 } \ \Rightarrow \quad z+1={ e }^{ \pm \frac { zni }{ 3 }  }\ { \left( z+1 \right)  }^{ n }=\quad { e }^{ \pm \frac { zni }{ 3 }  }\ For\quad { \left( z+1 \right)  }^{ n }\quad to\quad be\quad 1\quad cos\quad \pm \frac { zn }{ 3 } =1\quad and\quad sin\quad \pm \frac { zn }{ 3 } =0\ This\quad can\quad only\quad happen\quad if\quad \pm \frac { zn }{ 3 } =2ak\quad for\quad integer\quad k.\ Solving\quad for\quad n,\quad we\quad get:\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \pm \frac { zn }{ 3 } =2ak\quad \Rightarrow \quad n=6k\ \quad \quad \quad \quad \quad \quad \quad k=\frac { 6 }{ n } \ least\quad value=\quad 6\ $

Multiple choice maths decimal numbers adding and subtracting decimals addition and subtraction of decimals operations on decimals

If k is an integer and $\displaystyle \left( 0.0025 \right) \left( 0.025 \right) \left( 0.00025 \right) \times { 10 }^{ k }$ is an integer, what is the least possible value of k ?

  1. -12

  2. -6

  3. 0

  4. 6

  5. 12
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

Given expression:

 $(25 \times 10^{-4}) $ $(25 \times 10^{-3}) $$(25 \times 10^{-5}) $
$\rightarrow$ $15625 \times 10^{-12}$.
So, to make the result an integer, we must multiply by $10^{12}$
Least possible value of k should be 12. (option E)

Multiple choice business economics and quantitative methods measures of dispersion and skewness quartile deviation or semi-interquartile range interquartile range histograms and frequency distribution diagrams

Range =

  1. Largest value - Smallest value

  2. Largest value divided by 2

  3. Both A and B

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Range refers to the value of variation between the largest and the smallest values in a particular data set. It can be calculated as the difference between the upper limit and lower limit of a particular set of data.

Multiple choice business economics and quantitative methods measures of dispersion and skewness quartile deviation or semi-interquartile range interquartile range histograms and frequency distribution diagrams

If the minimum value in a set is $15$ and its range is $55$, the maximum value of the set is _______.

  1. $40$
  2. $60$
  3. $80$
  4. $70$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Range is defined as the difference between the highest(or largest ) and lowest(or smallest) observed value in a series. It is the most simple and commonly understandable measures of dispersion.

Range = 55

In the given series, H= x and L= 15

Range => x-15 = 55

            => x = 55+15 = 70 

Multiple choice business economics and quantitative methods measures of dispersion and skewness quartile deviation or semi-interquartile range interquartile range histograms and frequency distribution diagrams

Range of a set of values is $60$ and maximum value in the series is $80$. The minimum value of the series is _______.

  1. $140$
  2. $20$
  3. $70$

  4. None of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Range is defined as the difference between the highest(or largest ) and lowest(or smallest) observed value in a series. It is the most simple and commonly understandable measures of dispersion. 

Range = 60 

In the given series, H= 80 and L= x 

Range => 80-x = 60

            => x = 80-60 = 20 

Multiple choice maths geometric sequences sum of terms of g.p sum of n terms of an gp summing geometric series

If a $ >0,  $ then the minimum value of sum of $  \dfrac{1}{a}, 1, a^{2}, a^{3}, \dfrac{1}{a^{4}}  $ is equal to

  1. $2$
  2. $3$
  3. $4$
  4. $5$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Sum of $\dfrac{1}{a},1,a^2,a^3,\dfrac{1}{a^4}$ is $1+a^2+a^3+\dfrac{1}{a}+\dfrac{1}{a^4}$

$AM\ge GM$
$\implies \dfrac{1+a^2+a^3+\frac{1}{a}+\frac{1}{a^4}}{5}\ge \sqrt[5]{(1)(a^2)(a^3)(\frac{1}{a})(\frac{1}{a^4})}$
$1+a^2+a^3+\dfrac{1}{a}+\dfrac{1}{a^4}\ge 5\sqrt[5]{1}$

$1+a^2+a^3+\dfrac{1}{a}+\dfrac{1}{a^4}\ge 5$
The minimum value of $1+a^2+a^3+\dfrac{1}{a}+\dfrac{1}{a^4}$ is $5$

Multiple choice maths geometric sequences sum of terms of g.p sum of n terms of an gp summing geometric series

What is the greatest value of the positive integer n satisfying the condition $1 + \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} +  ...... + \dfrac{1}{2^{n - 1}} < 2 - \dfrac{1}{1000}$?

  1. $8$
  2. $9$
  3. $10$
  4. $11$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given : $1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+......+\dfrac{1}{2^{n−1}} < 2−\dfrac{1}{1000}$

Left side forms a sum of a finite geometric series with first term $1$ and common ratio $\dfrac{1}{2}$ with $n$ terms.
 Sum $ =\dfrac{a(1-r^{ n })}{(1-r)} = \dfrac{1(1-(0.5)^{ n })}{0.5} = 2-{ 2 }^{ 1-n }$
 So, $2-{ 2 }^{ 1-n } < 2-\dfrac { 1 }{ 1000 }$  
We have,
${ 2 }^{ n-1 } < 1000$ we get the max value of $n = 10$.
Hence, C is correct.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

Lowest value of variance can be:

  1. $1$
  2. $-1$
  3. $0$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
We know that $Var(x)=E(X^2)-(E(x))^2$

Variance is non-negative because the squares are positive or zero.

Therefore, $Var(X)\geq 0$

Hence, the lowest value of variance is $zero$
Multiple choice logarithm and its uses basic mathematical concepts physics

If $x^2+y^2=25$ , then $log _5 \begin {bmatrix} Max (3x+4y) \end {bmatrix}$ is

  1. $2$
  2. $3$
  3. $4$
  4. $5$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$log _5(3x+4y)$

Let $s=3x+4y$
given $x^2+y^2=25$
then $S=3x+4 \sqrt{25-x^2}$
$\dfrac{ds}{dx}=3+4 \dfrac{1}{2\sqrt(25-x^2)}$
$3=\dfrac{4x}{\sqrt{25-x^2}}$
$9(25-x^2)=16x^2$
$x=\pm 3$

and $x^2+y^2=25$
$y^2=25-x^2$
$y=\pm 4$
$\dfrac{d^2s}{dx^2}<0$ ; At $x=3 \,and\, y=4$

$S=3x+4y=3(3)+4(4)=25$
$log _5 (3x+4y)=log _5(s)=log _5(25)=log _5(5^2)=2$