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Let g: (0, ∞)  R be a differentiable function such that  for all x > 0, where c is an arbitrary constant. Then,
  • a)
    g is decreasing in (0, π/4)
  • b)
    g is increasing in (0, π/4)
  • c)
    g + g' is increasing in (0, π/2)
  • d)
    g + g' is decreasing in (0, π/2)
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
Let g: (0, ∞) R be a differentiable function such that for all x...

On differentiating both sides w.r.t. x, we get

(ex + 1) x (cos x - sin x) + g(x)(ex + 1 - xex)
= (ex + 1)(g(x) + x g'(x)) - ex . x . g(x)
⇒ g'(x) = cos x - sin x
⇒ g(x) = sin x + cos x + C
g(x) is increasing in (0, 
π
/4)
g'' (x) = -sin x - cos x < 0
⇒ g'(x) is decreasing function
let h(x) = g(x) + g' (x) = 2 cos x + C
⇒ h' (x) = g'(x) + g'' (x) = -2 sin x < 0
⇒ h is decreasing
let ϕ(x) = g(x) - g'(x) = 2 sin x + C
⇒ ϕ' (x) = g' (x) - g'' (x) = 2 cos x > 0
⇒ ϕ is increasing
Hence, option D is correct.
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Let g: (0, ∞) R be a differentiable function such that for all x > 0, where c is an arbitrary constant. Then,a)g is decreasing in (0,π/4)b)g is increasing in (0, π/4)c)g + g is increasing in(0, π/2)d)g + g is decreasing in (0, π/2)Correct answer is option 'D'. Can you explain this answer?
Question Description
Let g: (0, ∞) R be a differentiable function such that for all x > 0, where c is an arbitrary constant. Then,a)g is decreasing in (0,π/4)b)g is increasing in (0, π/4)c)g + g is increasing in(0, π/2)d)g + g is decreasing in (0, π/2)Correct answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let g: (0, ∞) R be a differentiable function such that for all x > 0, where c is an arbitrary constant. Then,a)g is decreasing in (0,π/4)b)g is increasing in (0, π/4)c)g + g is increasing in(0, π/2)d)g + g is decreasing in (0, π/2)Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let g: (0, ∞) R be a differentiable function such that for all x > 0, where c is an arbitrary constant. Then,a)g is decreasing in (0,π/4)b)g is increasing in (0, π/4)c)g + g is increasing in(0, π/2)d)g + g is decreasing in (0, π/2)Correct answer is option 'D'. Can you explain this answer?.
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