PYQSide ExamsMy practice

Application of Derivatives · Question 12

JEE Main 2026 (Online) 21st January Evening ShiftMCQMedium📈 Topic in 16% papers+4−1
1

Let f : R R be a twice differentiable function such that f''(x) > 0 for all x R and f'(a - 1) = 0, where a is a real number. Let g(x) = f( 2 x - 2 x + a), 0 < x < 2. Consider the following two statements: (I) g is increasing in (0, 4 ) (II) g is decreasing in ( 4, 2 ) Then,

  1. Both (I) and (II) are True
  2. Neither (I) nor (II) is True
  3. Only (I) is True
  4. Only (II) is True

Answer

(2) Neither (I) nor (II) is True

Solution

f''(x) > 0 ⇒ f' is strictly increasing, and f'(a - 1) = 0

Let u = 2 x - 2 x + a = ( x - 1)2 + a - 1 a - 1

f'(u) f'(a - 1) = 0, with f'(u) > 0 for x 4

g'(x) = f'(u) (2 x 2 x - 2 2 x) = 2 2 x ( x - 1) f'(u)

For 0 < x < 4: x - 1 < 0 ⇒ g'(x) < 0 ⇒ g is decreasing

For 4 < x < 2: x - 1 > 0 ⇒ g'(x) > 0 ⇒ g is increasing

(I) is false and (II) is false

Answer: (2)

JEE Main 2026 (Online) 21st January Evening Shift

Keep practising — Application of Derivatives

2

Let the function f(x) = x3 + 3x + 3, x 0 be strictly increasing in (- , 1) ( 2, ) and strictly decreasing in ( 3, 4) ( 4, 5). Then i=15 i2 is equal to

  1. 48
  2. 40
  3. 36
  4. 28
JEE Main 2025 (Online) 8th April Evening Shift
3

Let x = -1 and x = 2 be the critical points of the function f(x) = x3 + ax2 + b e |x| + 1, x 0. Let m and M respectively be the absolute minimum and the absolute maximum values of f in the interval [-2, - 12 ]. Then |M + m| is equal to (Take e 2 = 0.7):

  1. 21.1
  2. 19.8
  3. 22.1
  4. 20.9
JEE Main 2025 (Online) 7th April Morning Shift
4

Let f : R R be a polynomial function of degree four having extreme values at x = 4 and x = 5. If x 0 f(x)x2 = 5, then f(2) is equal to :

  1. 8
  2. 10
  3. 12
  4. 14
JEE Main 2025 (Online) 7th April Evening Shift
5

Let a > 0. If the function f(x) = 6x3 - 45ax2 + 108a2x + 1 attains its local maximum and minimum values at the points x1 and x2 respectively such that x1 x2 = 54, then a + x1 + x2 is equal to :

  1. 15
  2. 13
  3. 24
  4. 18
JEE Main 2025 (Online) 4th April Evening Shift
6

Let f : R R be a function defined by f(x) = ||x + 2| - 2|x| |. If m is the number of points of local minima and n is the number of points of local maxima of f, then m + n is

  1. 3
  2. 4
  3. 2
  4. 5
JEE Main 2025 (Online) 3rd April Evening Shift
7

If the function f(x) = 2x3 - 9ax2 + 12a2x + 1, where a > 0, attains its local maximum and local minimum values at p and q, respectively, such that p2 = q, then f(3) is equal to :

  1. 55
  2. 37
  3. 10
  4. 23
JEE Main 2025 (Online) 2nd April Morning Shift
8

The sum of all local minimum values of the function f(x) = cases 1 - 2x, & x < -1 \ 13(7 + 2|x|), & -1 x 2 \ 1118(x - 4)(x - 5), & x > 2 cases is

  1. 16772
  2. 15772
  3. 17172
  4. 13172
JEE Main 2025 (Online) 28th January Morning Shift

All Application of Derivatives PYQs →

PDFDownload Application of Derivatives PYQs as PDF Free, with answer key and a QR code to every solution — print it or study offline⬇
All JEE Main Mathematics chapters
Class 11SetsRelations and FunctionsTrigonometric FunctionsComplex Numbers and Quadratic EquationsPermutations and CombinationsBinomial TheoremSequences and SeriesStraight LinesConic SectionsIntroduction to Three Dimensional GeometryLimits and DerivativesStatisticsProbability
Class 12Relations and Functions (Class 12)Inverse Trigonometric FunctionsMatricesDeterminantsContinuity and DifferentiabilityApplication of DerivativesIntegralsApplication of IntegralsDifferential EquationsVector AlgebraThree Dimensional GeometryProbability (Class 12)