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Indian Navy MR Application of Derivatives — practice questions

Explore 30 free Application of Derivatives questions in Indian Navy MR 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 1–30 of 30.

The **slope of the tangent** to the curve $y = f(x)$ at point $(a, f(a))$ is:Read the answer and worked solution →If $f(x) = x^x$ for $x > 0$, then $f$ has a minimum at $x = $:Read the answer and worked solution →For f(x) = x³ - 3x² + 4, the function is increasing on intervals:Read the answer and worked solution →Concavity test: f(x) = e^x is:Read the answer and worked solution →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 →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

The **slope of the tangent** to the curve $y = f(x)$ at point $(a, f(a))$ is:

  1. $f(a)$
  2. $f'(a)$
  3. $1/f'(a)$
  4. $f''(a)$
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Correct answer: B. $f'(a)$

Tangent slope at $x = a$ equals $f'(a)$, the derivative evaluated at that point. Normal slope = $-1/f'(a)$ (perpendicular).

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If $f(x) = x^x$ for $x > 0$, then $f$ has a minimum at $x = $:

  1. 1
  2. $1/e$ (minimum value = $e^{-1/e}$)
  3. 0
  4. $e$
Show answer and solution

Correct answer: B. $1/e$ (minimum value = $e^{-1/e}$)

Take $\ln f = x \ln x$. **Differentiate:** $f'/f = \ln x + 1 = 0$ ⇒ $\ln x = -1$ ⇒ $x = 1/e$. $f''$ check confirms min. Value: $f(1/e) = (1/e)^{1/e} = e^{-1/e}$.

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For f(x) = x³ - 3x² + 4, the function is increasing on intervals:

  1. (-∞, 0)
  2. (2, ∞)
  3. (0, 2) only
  4. (-∞, 0) ∪ (2, ∞) (where f'(x) = 3x(x-2) > 0)
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Correct answer: D. (-∞, 0) ∪ (2, ∞) (where f'(x) = 3x(x-2) > 0)

We have f'(x) = 3x² - 6x. So f'(x) = 3x(x - 2). f' > 0 when x < 0 OR x > 2 (increasing). f' < 0 on (0, 2) (decreasing). Local max at x = 0, local min at x = 2.

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