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Indian Navy AA (Artificer Apprentice) Optimization — practice questions

Explore 26 free Optimization questions in Indian Navy AA (Artificer Apprentice) Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Optimization questions and exercises with answers

Showing questions 1–26 of 26.

A rectangle is inscribed in a semicircle of radius $R$ with base on the diameter. The **maximum area** is:Read the answer and worked solution →What does the sign of f ′(x) tell you?Read the answer and worked solution →Complete the definition of a critical pointRead the answer and worked solution →Order the first derivative testRead the answer and worked solution →Match each sign change to its verdictRead the answer and worked solution →The second derivative testRead the answer and worked solution →True or false: maxima and minimaRead the answer and worked solution →Order the working rule for absolute maxima and minimaRead the answer and worked solution →Tap every point you must test for an absolute maximumRead the answer and worked solution →Local or absolute?Read the answer and worked solution →Complete the rate of change statementRead the answer and worked solution →The chain rule for related ratesRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →True or false: increasing and decreasingRead the answer and worked solution →Flip and pair: symbol with meaningRead the answer and worked solution →Match each situation to the right toolRead the answer and worked solution →Tap every statement true of a local maximum at cRead the answer and worked solution →Reading the tests backRead the answer and worked solution →Which test is being described?Read the answer and worked solution →Assertion and reason: rates of changeRead the answer and worked solution →Spot the odd oneRead the answer and worked solution →Complete the second derivative testRead the answer and worked solution →True or false: scope of this chapterRead the answer and worked solution →

Worked practice questions

A rectangle is inscribed in a semicircle of radius $R$ with base on the diameter. The **maximum area** is:

  1. $R^2$ (when the rectangle is a square inscribed)
  2. $2R^2$
  3. $R^2/2$
  4. $\pi R^2/2$
Show answer and solution

Correct answer: A. $R^2$ (when the rectangle is a square inscribed)

Half-width = $x$, height $y = \sqrt{R^2 - x^2}$. Area $A = 2xy = 2x\sqrt{R^2 - x^2}$. We have $dA/dx = 2\sqrt{R^2 - x^2} - 2x^2/\sqrt{R^2 - x^2}$. So $dA/dx = 0$. ⇒ $R^2 = 2x^2$ ⇒ $x = R/\sqrt 2$. Then $y = R/\sqrt 2$, area = $2 \cdot (R/\sqrt 2)(R/\sqrt 2) = R^2$.

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What does the sign of f ′(x) tell you?

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Items
  1. Item
    f ′(x) > 0 throughout
    Category
    Increasing
  2. Item
    x₁ < x₂ gives f(x₁) < f(x₂)
    Category
    Increasing
  3. Item
    f ′(x) < 0 throughout
    Category
    Decreasing
  4. Item
    x₁ < x₂ gives f(x₁) > f(x₂)
    Category
    Decreasing
  5. Item
    f(x) = c for all x
    Category
    Constant
  6. Item
    f ′(x) = 0 throughout the interval
    Category
    Constant
Categories
  1. Item
    Increasing
  2. Item
    Decreasing
  3. Item
    Constant
Your syllabus gives f ′(x) ≥ 0 on an interval as the test for increasing and f ′(x) ≤ 0 for decreasing. Everything else in the chapter builds on reading that sign. Source: syllabus Maths Part I, Ch 6 Application of Derivatives.

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Complete the definition of a critical point

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Missing parts
  1. Answer
    0
    Choices
    1. 0
    2. 1
    3. undefined
  2. Answer
    not differentiable
    Choices
    1. not differentiable
    2. not continuous
    3. not defined
Complete the statement
A point c in the domain of f is a critical point if either f ′(c) = {0}, or f is {1} at c.
Your syllabus's definition covers two cases, and the second one is the one people forget. Source: syllabus Maths Part I, Ch 6 Application of Derivatives.

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