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 comparison of irrational numbers surds and law of surds rational and irrational numbers exponents maths

Which of the following numbers is the least ?
$\displaystyle (0.5)^{2},\sqrt{0.49},\sqrt[3]{0.008},0.23$

  1. $\displaystyle (0.5)^{2}$
  2. $\displaystyle \sqrt{0.49}$
  3. $\displaystyle \sqrt[3]{0.008}$

  4. 0.23
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$ (0.5)^{2}=0.25$
$\sqrt{0.49}=0.7;$
$ \sqrt[3]{0.008}=\sqrt[3]{.2^3}=0.2$
$0.23$
Arranging in ascending order the numbers are $0.2< 0.23< 0.25< 0.7$
$ \therefore \sqrt[3]{0.008}=0.2$ is the least

Multiple choice comparison of irrational numbers surds and law of surds rational and irrational numbers exponents maths

The smallest of $\displaystyle \sqrt{8}+\sqrt{5},\sqrt{7}+\sqrt{6},\sqrt{10}+\sqrt{3}$ and $\displaystyle \sqrt{11}+\sqrt{2}$ is 

  1. $\displaystyle \sqrt{8}+\sqrt{5}$
  2. $\displaystyle \sqrt{7}+\sqrt{6}$
  3. $\displaystyle \sqrt{10}+\sqrt{3}$
  4. $\displaystyle \sqrt{11}+\sqrt{2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\displaystyle \sqrt{8}+\sqrt{5}=2.83+2.24=5.07$
$\displaystyle \sqrt{7}+\sqrt{6}=2.65+2.45=5.09$
$\displaystyle \sqrt{10}+\sqrt{13}=3.16+3.61=6.77$
$\displaystyle \sqrt{11}+\sqrt{12}=3.32+1.41=4.73$
$\displaystyle \therefore $ Smallest is $\displaystyle \sqrt{11}+\sqrt{2}$

Multiple choice comparison of irrational numbers surds and law of surds rational and irrational numbers exponents maths

Which of the following is smallest?

  1. $\sqrt [4]{5}$
  2. $\sqrt [5]{4}$
  3. $\sqrt {4}$
  4. $\sqrt {3}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let us rewrite the given set of magnitudes $\sqrt [ 4 ]{ 5 } ,\sqrt [ 5 ]{ 4 } ,\sqrt { 4 },\sqrt {3}$ as follows:


$\sqrt [ 4 ]{ 5 } ={ \left( 5 \right)  }^{ \dfrac { 1 }{ 4 }  }\ \sqrt [ 5 ]{ 4 } ={ \left( 4 \right)  }^{ \dfrac { 1 }{ 5 }  }\ \sqrt { 4 } ={ \left( 4 \right)  }^{ \dfrac { 1 }{ 2 }  }\ \sqrt { 3 } ={ \left( 3 \right)  }^{ \dfrac { 1 }{ 2 }  }$

 
We now take the LCM of the denominators of the powers to make the denominators same, then the above magnitudes will be:

$\sqrt [ 4 ]{ 5 } ={ \left( 5 \right)  }^{ \dfrac { 1\times 5 }{ 4\times 5 }  }={ \left( 5 \right)  }^{ \dfrac { 5 }{ 20 }  }={ \left( { 5 }^{ 5 } \right)  }^{ \dfrac { 1 }{ 20 }  }={ \left( 3125 \right)  }^{ \dfrac { 1 }{ 20 }  }=\sqrt [ 20 ]{ 3125 } \\ \sqrt [ 5 ]{ 4 } ={ \left( 4 \right)  }^{ \dfrac { 1\times 4 }{ 5\times 4 }  }={ \left( 4 \right)  }^{ \dfrac { 4 }{ 20 }  }={ \left( { 4 }^{ 4 } \right)  }^{ \dfrac { 1 }{ 20 }  }={ \left( 256 \right)  }^{ \dfrac { 1 }{ 20 }  }=\sqrt [ 20 ]{ 256 } \\ \sqrt { 4 } ={ \left( 4 \right)  }^{ \dfrac { 1\times 10 }{ 2\times 10 }  }={ \left( 4 \right)  }^{ \dfrac { 10 }{ 20 }  }={ \left( { 4 }^{ 10 } \right)  }^{ \dfrac { 1 }{ 20 }  }={ \left( 1048576 \right)  }^{ \dfrac { 1 }{ 20 }  }=\sqrt [ 20 ]{ 1048576 } \\ \sqrt { 3 } ={ \left( 3 \right)  }^{ \dfrac { 1\times 10 }{ 2\times 10 }  }={ \left( 3 \right)  }^{ \dfrac { 10 }{ 20 }  }={ \left( { 3 }^{ 10 } \right)  }^{ \dfrac { 1 }{ 20 }  }={ \left( 2187 \right)  }^{ \dfrac { 1 }{ 20 }  }=\sqrt [ 20 ]{ 2187 }$     

Now, the descending order is as shown below:

$\sqrt [ 20 ]{ 1048576 } >\sqrt [ 20 ]{ 3125 } >\sqrt [ 20 ]{ 2187 } >\sqrt [ 20 ]{ 256 } \\ \Rightarrow \sqrt { 4 } >\sqrt [ 4 ]{ 5 } >\sqrt { 3 } >\sqrt [ 5 ]{ 4 }$ 

Hence, the smallest magnitude is $\sqrt [ 5 ]{ 4 }$.
Multiple choice

Simplicity is?

  1. Reducing components of composition to only what is necessary

  2. Finding the lowest denominator

  3. Using the lowest f-stop possible

  4. Using the mode dial on Tv
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Simplicity in composition involves removing unnecessary elements to focus the viewer's attention on the subject.

Multiple choice maths average arithmetic mean of ap introduction to averages means
If $a+b+c+d+e+f=12$ then the maximum value of $ab+bc+cd+de+ef+fa$ is (a, b, c, d, e, f are non negative real numbers)
  1. $36$
  2. $24$
  3. $30$

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
AM $\ge$ GM
$\cfrac { a+b+c+d+e+f }{ 6 } \ge { (abcdef) }^{ 1/6 }\\ { 2 }^{ 2 }\ge { (abcdef) }^{ 1/3 }\quad \quad \quad (1)\\ \cfrac { ab+bc+cd+de+ef+fa }{ 6 } \ge { (abcdef) }^{ 2/6 }\\ ab+bc+cd+de+ef+fa\ge 6{ (abcdef) }^{ 1/3 } \quad (2)$
Dividing $(2)$ by $(1)$
$ab+bc+cd+de+ef+fa \ge 24$
Multiple choice

Consider the following Integer Programming problem: Maximize z = 2x + 3y subject to x + y ≤ 5, x, y ≥ 0, x, y ∈ Z. What is the optimal solution to this problem?

  1. x = 2, y = 3

  2. x = 3, y = 2

  3. x = 4, y = 1

  4. x = 5, y = 0
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To solve this problem, you can use a branch-and-bound algorithm or a dynamic programming approach. The optimal solution is x = 2, y = 3, which gives an objective value of z = 12.

Multiple choice

What is the minimum acceptable value of Cronbach's alpha coefficient for a language test?

  1. 0.50

  2. 0.60

  3. 0.70

  4. 0.80
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The minimum acceptable value of Cronbach's alpha coefficient for a language test is generally considered to be 0.70. However, this value may vary depending on the purpose of the test and the number of items in the test.

Multiple choice

Consider the inequality -3x + 4 > 7. What is the smallest integer value of x that satisfies the inequality?

  1. -1

  2. 0

  3. 1

  4. 2
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

To solve the inequality -3x + 4 > 7, we need to isolate x. Subtracting 4 from both sides, we get -3x > 3. Dividing both sides by -3 (reversing the inequality since we are dividing by a negative number), we get x < -1. The smallest integer value that satisfies this inequality is -2.

Multiple choice

In the inequality 3x - 2 > 8, what is the smallest integer value of x that satisfies the inequality?

  1. 3

  2. 4

  3. 5

  4. 6
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

To solve the inequality 3x - 2 > 8, we need to isolate x. Adding 2 to both sides, we get 3x > 10. Dividing both sides by 3, we get x > 10/3. The smallest integer value that satisfies this inequality is 4.

Multiple choice

What is the name of the theorem that states that if a function is continuous on a closed interval, then it attains both a maximum and a minimum value on that interval?

  1. Extreme Value Theorem

  2. Rolle's Theorem

  3. Mean Value Theorem

  4. Cauchy's Integral Theorem
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Extreme Value Theorem states that if a function is continuous on a closed interval, then it attains both a maximum and a minimum value on that interval.

Multiple choice

Let $a, b, c$ be positive real numbers such that $a + b + c = 3$. Find the maximum value of the expression $a^2 + b^2 + c^2$.

  1. 3

  2. 6

  3. 9

  4. 12
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The maximum value of $a^2 + b^2 + c^2$ is achieved when $a = b = c = 1$. Therefore, the maximum value is $1^2 + 1^2 + 1^2 = 3.