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Application of Derivatives

JEE Main 2024 (Online) 29th January Morning ShiftNumericalMedium📈 Topic in 3% papers+4
Ques. 1JEE 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 .

Answer

5

Solution

f(x)=0 2x=x2: x = 2, x = 4, and one root in (−1, 0), as f(-1)=- 12<0, f(0)=1>0

f'(x)=0 2x 2=2x: one root in (0, 1), one in (1, 4), as f'(0)>0, f'(1)=2 2-2<0, f'(4)=16 2-8>0

[figure]

∵ f''(x)=2x( 2)2-2 has only one zero ⇒ f' has at most 2 zeros ⇒ n = 2

⇒ by Rolle's theorem f has at most 3 zeros ⇒ m = 3

∴ m + n = 5

Answer: 5

Keep practising — Application of Derivatives

Ques. 2JEE 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. 3JEE 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. 4JEE 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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