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Indian Navy INET Application of Derivatives — practice questions

Explore 27 free Application of Derivatives questions in Indian Navy INET Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Application of Derivatives questions and exercises with answers

Showing questions 1–27 of 27.

By the second derivative test, if $f'(c) = 0$ and $f''(c) > 0$, then $c$ is a point of:Read the answer and worked solution →The critical points of f(x) = x³ − 3x + 1 are found by solving f'(x) = 0: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

By the second derivative test, if $f'(c) = 0$ and $f''(c) > 0$, then $c$ is a point of:

  1. inflection
  2. local minimum
  3. local maximum
  4. discontinuity
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Correct answer: B. local minimum

f'(c)=0 with f''(c)>0 indicates a local minimum (curve concave up).

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The critical points of f(x) = x³ − 3x + 1 are found by solving f'(x) = 0:

  1. $x = ±1$ (from $3x^2 - 3 = 0$)
  2. $x = 0$ only (single critical)
  3. $x = ±2$ (incorrect calculation)
  4. $x = ±3$ (mistaken derivation)
Show answer and solution

Correct answer: A. $x = ±1$ (from $3x^2 - 3 = 0$)

We have f'(x) = 3x² − 3. That is f'(x) = 0 → x². So f'(x) = 1. → x = ±1. Critical points at x = 1 (local min, f''(1) = 6 > 0) + x = -1 (local max, f''(-1) = -6 < 0).

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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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