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Application of Derivatives

JEE Main 2024 (Online) 30th January Morning ShiftMCQMedium📈 Topic in 3% papers+4−1
Ques. 1JEE Main 2024, 30 January (Morning Shift)

Let g:R be a non constant twice differentiable function such that g' ( 12 )=g' ( 32 ). If a real valued function f is defined as f(x)= 12[g(x)+g(2-x)], then

  1. f''(x)=0 for at least two x in (0,2)
  2. f' ( 32 )+f' ( 12 )=1
  3. f''(x)=0 for no x in (0,1)
  4. f''(x)=0 for exactly one x in (0,1)

Answer

(1) f''(x)=0 for at least two x in (0,2)

Solution

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

f' ( 12 )= g' ( 12 )-g' ( 32 )2=0

f' ( 32 )= g' ( 32 )-g' ( 12 )2=0

f'(1)= g'(1)-g'(1)2=0

f' vanishes at 12, 1, 32, Rolle's theorem on f' in [ 12,1 ] and [1, 32 ]

f''(c1)=0, c1 ( 12,1 ) and f''(c2)=0, c2 (1, 32 )

(A): two roots c1, c2 in (0,2) ⇒ (A) is true

(B): f' ( 32 )+f' ( 12 )=0 1 ⇒ (B) is false

(C): c1 (0,1) is a root ⇒ (C) is false

(D): only "at least one" root in (0,1) is guaranteed ⇒ (D) is not certain

Answer: (1)

Keep practising — Application of Derivatives

Ques. 2JEE Main 2024, 30 January (Morning Shift)

The maximum area of a triangle whose one vertex is at (0,0) and the other two vertices lie on the curve y=-2x2+54 at points (x, y) and (-x, y), where y>0, is :

  1. 108
  2. 122
  3. 88
  4. 92
Ques. 3JEE Main 2024, 30 January (Evening Shift)

Let f(x)=(x+3)2(x-2)3, x [-4,4]. If M and m are the maximum and minimum values of f, respectively in [-4,4], then the value of M-m is

  1. 108
  2. 392
  3. 608
  4. 600
Ques. 4JEE Main 2024, 29 January (Morning Shift)

In an A.P., the sixth term a6=2. If the product a1a4a5 is the greatest, then the common difference of the A.P. is equal to

  1. 23
  2. 58
  3. 32
  4. 85
Ques. 5JEE Main 2024, 29 January (Morning Shift)

Consider the function f: [ 12,1 ] R defined by f(x)=4 2x3-3 2x-1. Consider the statements (I) The curve y=f(x) intersects the x-axis exactly at one point. (II) The curve y=f(x) intersects the x-axis at x= 12. Then

  1. Both (I) and (II) are correct.
  2. Only (I) is correct.
  3. Both (I) and (II) are incorrect.
  4. Only (II) is correct.
Ques. 6JEE Main 2024, 29 January (Morning Shift)

Let f(x)=2x-x2,x R. If m and n are respectively the number of points at which the curves y=f(x) and y=f'(x) intersect the x-axis, then the value of m+n is .

Ques. 7JEE Main 2024, 29 January (Evening Shift)

The function f(x)= xx2-6x-16, x R-\-2,8\

  1. decreases in (- ,-2) (-2,8) (8, )
  2. increases in (- ,-2) (-2,8) (8, )
  3. decreases in (-2,8) and increases in (- ,-2) (8, )
  4. decreases in (- ,-2) and increases in (8, )
Ques. 8JEE Main 2024, 29 January (Evening Shift)

The function f(x)=2x+3(x) 23, x R, has

  1. exactly one point of local minima and no point of local maxima
  2. exactly one point of local maxima and exactly one point of local minima
  3. exactly two points of local maxima and exactly one point of local minima
  4. exactly one point of local maxima and no point of local minima
Ques. 9JEE Main 2024, 27 January (Evening Shift)

Let g(x)=3f ( x3 )+f(3-x) and f''(x)>0 for all x (0,3). If g is decreasing in (0, ) and increasing in ( ,3), then 8 is :

  1. 0
  2. 24
  3. 18
  4. 20

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