Verified Solution • 2025–26 Session moderate

QJEEM19Ma84ed: JEE Main 2019 Mathematics - Class 12 JEEM19Ma84ed Solution: Let ƒ : [0, 2] → R be a twice differentiable function such that …

Let ƒ : [0, 2] → R be a twice differentiable function such that ƒ''(x) > 0, for all x ∈ (0, 2). If ϕ(x) = ƒ(x) + ƒ(2 – x), then ϕ is :

Multiple Choice Options

  • A decreasing on (0, 2)
  • B decreasing on (0, 1) and increasing on (1, 2)
  • C increasing on (0, 2)
  • D increasing on (0, 1) and decreasing on (1, 2)

Step-by-Step Verified Solution

ϕ(x) = ƒ(x) + ƒ(2 – x)

⇒ ϕ'(x) = ƒ'(x) - ƒ'(2 – x)

Since ƒ''(x) > 0 for all x ∈ (0, 2)

⇒ ƒ'(x) is an increasing function for all x ∈ (0, 2).

Case 1 : When ϕ(x) is increasing function

So ϕ'(x) > 0

⇒ ƒ'(x) - ƒ'(2 – x) > 0

⇒ ƒ'(x) > ƒ'(2 – x)

⇒ x > 2 – x

⇒ x > 1

∴ ϕ(x) is increasing on (1, 2).

Case 2 : When ϕ(x) is decreasing function

So ϕ'(x) < 0

⇒ ƒ'(x) - ƒ'(2 – x) < 0

⇒ ƒ'(x) < ƒ'(2 – x)

⇒ x < 2 – x

⇒ x < 1

∴ ϕ(x) is decreasing on (0, 1).

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