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Continuity and Differentiability

JEE Main 2024 (Online) 27th January Morning ShiftNumericalMedium📈 Topic in 12% papers+4
Ques. 1JEE Main 2024, 27 January (Morning Shift)

Let f(x)=x3+x2 f'(1)+x f''(2)+f'''(3), x . Then f'(10) is equal to .

Answer

202

Solution

∵ f'(1), f''(2), f'''(3) are constants:

f'(x)=3x2+2xf'(1)+f''(2), f''(x)=6x+2f'(1), f'''(x)=6

f''(2)=12+2f'(1)

f'(1)=3+2f'(1)+f''(2)=15+4f'(1) ⇒ f'(1)=-5, f''(2)=2

⇒ f'(x)=3x2-10x+2

∴ f'(10)=300-100+2=202

Answer: 202

Keep practising — Continuity and Differentiability

Ques. 2JEE Main 2024, 27 January (Morning Shift)

Let for a differentiable function f:(0, ) , f(x)-f(y) e ( xy )+x-y, x, y (0, ). Then n=120f' ( 1n2 ) is equal to .

Ques. 3JEE Main 2024, 27 January (Evening Shift)

Consider the function f:(0,2) defined by f(x)= x2+ 2x and the function g(x) defined by g(x)= cases \f(t)\, & 0<t x and 0<x 1\ 32+x, & 1<x<2 cases. Then,

  1. g is continuous but not differentiable at x=1
  2. g is continuous and differentiable for all x (0,2)
  3. g is not continuous for all x (0,2)
  4. g is neither continuous nor differentiable at x=1

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