Mathematics

Maxima and Minima

129 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

The smallest positive integer x, for which 945x2 = N3, where N is a natural number, is

  1. 49

  2. 25

  3. 35

  4. 9
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

945 = 3³ × 5 × 7. For 945x² to be a perfect cube N³, x² must provide the missing factors to make all exponents multiples of 3. We need one more 3, one more 5, and two more 7s. So x² = 3 × 5 × 7² = 735, giving x = √735 = 35. Verification: 945 × 35² = 945 × 1225 = 1157625, and 1157625^(1/3) = 105.

Multiple choice

When TP reaches its maximum level, MP is

  1. positive

  2. negative

  3. zero

  4. the highest
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

When TP reaches its maximum level, MP is zero. MP becomes negative after this stage and MP is positive before this stage.

Multiple choice

When TP reaches its maximum level, MP is

  1. positive

  2. negative

  3. zero

  4. the highest
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

When TP reaches its maximum level, MP is zero. MP becomes negative after this stage and MP is positive before this stage.

Multiple choice

Given a function: (x, y) = 4x2 + 6y2 − 8x − 4y + 8 The optimal value of f(x, y)

  1. is a minimum equal to 10/3

  2. is a maximum equal to 10/3

  3. is a minimum equal to 8/3

  4. is a maximum equal to 8/3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Find partial derivatives: df/dx = 8x - 8 = 0 => x = 1. df/dy = 12y - 4 = 0 => y = 1/3. The second derivative test (f_xx = 8, f_yy = 12, f_xy = 0) shows a minimum since f_xx > 0 and D = f_xx*f_yy - f_xy^2 = 96 > 0. Value at (1, 1/3) is 4(1)^2 + 6(1/3)^2 - 8(1) - 4(1/3) + 8 = 4 + 2/3 - 8 - 4/3 + 8 = 4 - 2/3 = 10/3.

Multiple choice

The function f(x) = 2x2 - 3x2 - 36x + 2 has its maxima at

  1. x = - 2 only

  2. x = 0 only

  3. x = 3 only

  4. both x = - 2 and x = 3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

f(x) = 2x^3 - 3x^2 - 36x + 2. f'(x) = 6x^2 - 6x - 36. Setting f'(x) = 0 gives x^2 - x - 6 = 0, so (x-3)(x+2) = 0. Roots are x = 3 and x = -2. f''(x) = 12x - 6. f''(-2) = -24 - 6 = -30 (<0, maxima). f''(3) = 36 - 6 = 30 (>0, minima). Thus, maxima at x = -2.

Multiple choice business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii

$y = 48x - 2x^2$
where, $y=$ Total revenue $ $ $.
$x = $ Output
At what output is total revenue a maximum?

  1. $2$
  2. $12$
  3. $48$
  4. $4$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\Rightarrow$  We have given, $y=48x-2x^2$

$\Rightarrow$  Differentiate on both sides w.r.t $x$ we get,
$\Rightarrow$  $\dfrac{dy}{dx}=48-4x$          --- ( 1 )
$\Rightarrow$  Let $\dfrac{dy}{dx}=0$
$\Rightarrow$  $48-4x=0$
$\Rightarrow$  $4x=48$
$\Rightarrow$  $x=12$ is a turning point.
$\Rightarrow$  $\dfrac{d^2y}{dx^2}=-4$         [Differentiate ( 1 ) on both sides]
$\Rightarrow$  So, the turning point is a maximum 

Multiple choice maths decimal numbers comparing and ordering of decimals more or less comparing decimals

Of the numbers 0.16, $\displaystyle \sqrt{0.16}$, $\displaystyle (0.16)^{2}$ and $\displaystyle 0.1\overline{6}$ the least number is

  1. $\displaystyle (0.16)^{2}$
  2. $\displaystyle \sqrt{0.16}$

  3. 0.16
  4. $\displaystyle 0.1\overline{6}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

0.16 = 0.16
$\displaystyle \sqrt{0.16}=0.4 $  $\displaystyle \Rightarrow 0.0256=\left ( 0.16 \right )^{2}$
$\displaystyle \left ( 0.16 \right )^{2}=0.0256 $  $\displaystyle \Rightarrow 0.0256=\left ( 0.16 \right )^{2}$
$\displaystyle 0.\overline{16}=0.1666....$  $\displaystyle \Rightarrow 0.0256=\left ( 0.16 \right )^{2}$
is the least

Multiple choice maths fractions and decimals comparing and ordering of decimals more or less comparing decimals

The least number among $\frac { 4 }{ 9 } ,\sqrt { \frac { 9 }{ 49 }  } ,0.45\quad and\quad { (0.8) }^{ 2 }$ is

  1. $\frac 49$
  2. $\sqrt { \frac { 9 }{ 49 } }$

  3. 0.45
  4. $(0.8)^2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Convert all numbers to decimals or common bases for easy comparison. 4/9 is approximately 0.444, the square root of 9/49 is 3/7, which is approximately 0.428, 0.45 remains 0.45, and (0.8)^2 is 0.64. Comparing these decimal values, 3/7 (or sqrt(9/49)) is the smallest at approximately 0.428.

Multiple choice range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

The highest score of a certain data exceeds in lowest score by $16$ and coefficient of range is $\cfrac{1}{3}$. The sum of the highest score and the lowest score is

  1. $36$
  2. $48$
  3. $24$
  4. $18$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the highest score be $x _{m}$ and 

the lowest score be $x _{0}$
Given that highest score exceeds lowest score by $16$
$\implies x _m=x _0+16\implies x _m-x _0=16$ ————(1)

Coefficient of range is given by $\dfrac{x _m-x _0}{x _m+x _0}$

Given that coefficient of range is $\dfrac 13$

$\implies \dfrac{x _m-x _0}{x _m+x _0}=\dfrac 13$ ———(2)

Substitute (1) in (2) we get

$\dfrac{16}{x _m+x _0}=\dfrac 13$

$\implies x _m+x _0=16\times3=48$

Therefore sum of the highest score and lowest score is $48$

Multiple choice maths addition of vectors vectors from a geometric viewpoint basic concepts of vector introduction to vectors

If $a,b,c$ are unit vectors, then the maximum value of $|a+2b|^{2}+|b+3c|^{2}+|c+4a|^{2}$ is 

  1. $50$
  2. $21$
  3. $48$
  4. $58$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Given,

$\left|a+2b\right|^2+\left|b+3c\right|^2+\left|c+4a\right|^2$

$=\left(a+2b\right)^2+\left(b+3c\right)^2+\left(4a+c\right)^2$

$=a^2+4ab+4b^2+b^2+6bc+9c^2+c^2+8ac+16a^2$

$=17a^2+4ab+8ac+5b^2+10c^2+6bc$

$=50$