Start learning →

MP Board Class 12 (MPBSE) Application of Derivatives — practice questions

Explore 33 free Application of Derivatives questions in MP Board Class 12 (MPBSE) Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

Practise Application of Derivatives with Quill →

Application of Derivatives questions and exercises with answers

Showing questions 1–30 of 33.

A function f(x) is INCREASING on an interval if for all x in the interval:Read the answer and worked solution →The tangent to y = x² at the point (1, 1) has slope:Read the answer and worked solution →Rolle's theorem requires that on [a, b], in addition to continuity + differentiability:Read the answer and worked solution →At a critical point c where f'(c) = 0, if f''(c) > 0, then c is a:Read the answer and worked solution →To find the dimensions of a rectangle with maximum area + perimeter 20, set up: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 →For a particle moving along x(t) = t³ − 3t² + 5, the velocity at t = 2 is:Read the answer and worked solution →Fill the calculus turning point ruleRead 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 →

Worked practice questions

A function f(x) is INCREASING on an interval if for all x in the interval:

  1. $f''(x) > 0$ (positive second derivative)
  2. $f'(x) < 0$ (negative first derivative)
  3. $f'(x) > 0$ (positive first derivative)
  4. $f(x) > 0$ (positive function value)
Show answer and solution

Correct answer: C. $f'(x) > 0$ (positive first derivative)

f is INCREASING on an interval if f'(x) > 0 there. f''(x) > 0 means concave up (different concept). f' < 0 means decreasing.

Open this question →

The tangent to y = x² at the point (1, 1) has slope:

  1. $1$ (the y-value at x = 1)
  2. $2$ (since dy/dx = 2x, at x = 1)
  3. $0.5$ (incorrect inverse calculation)
  4. $0$ (a horizontal tangent)
Show answer and solution

Correct answer: B. $2$ (since dy/dx = 2x, at x = 1)

dy/dx = 2x. At x = 1: slope = 2(1) = 2. Tangent line: y − 1 = 2(x − 1) → y = 2x − 1. Normal slope = -1/2.

Open this question →

Rolle's theorem requires that on [a, b], in addition to continuity + differentiability:

  1. f'(a) = f'(b) (equal derivatives at ends)
  2. f''(a) = 0 = f''(b) (zero second derivative)
  3. f(a) + f(b) = 0 (sum is zero)
  4. f(a) = f(b) (function takes same value at endpoints)
Show answer and solution

Correct answer: D. f(a) = f(b) (function takes same value at endpoints)

**Rolle's theorem:** f continuous on [a,b], differentiable on (a,b), AND f(a) = f(b). **Conclusion:** there exists c in (a,b) with f'(c) = 0. Mean Value Theorem is similar but doesn't require f(a) = f(b).

Open this question →

How to practise

  1. Choose a Application of Derivatives question and try to work out your answer.
  2. Open the question to check the correct answer and read its worked solution.
  3. Continue in the app for bite-sized lessons, explanations and review with Quill. You can start without creating an account.

More Mathematics topics

Vectors & 3D GeometryBinomial TheoremGeometryPermutations and CombinationsProbability & StatisticsSequencesSequences and SeriesThree Dimensional Geometry