JEE Main 2019 — Application Of Derivatives Question with Solution

$$\varphi$$(x) = ƒ(x) + ƒ(2 – x)

$$\Rightarrow$$ $$\varphi$$'(x) = ƒ'(x) - ƒ'(2 – x)

Since ƒ''(x) > 0 for all x $$\in$$ (0, 2)

$$\Rightarrow$$ ƒ'(x) is an increasing function for all x $$\in$$ (0, 2).

Case 1 : When $$\varphi$$(x) is increasing function

So $$\varphi$$'(x) > 0

$$\Rightarrow$$ ƒ'(x) - ƒ'(2 – x) > 0

$$\Rightarrow$$ ƒ'(x) > ƒ'(2 – x)

$$\Rightarrow$$ x > 2 – x

$$\Rightarrow$$ x > 1

$$\therefore$$ $$\varphi$$(x) is increasing on (1, 2).

Case 2 : When $$\varphi$$(x) is decreasing function

So $$\varphi$$'(x) < 0

$$\Rightarrow$$ ƒ'(x) - ƒ'(2 – x) < 0

$$\Rightarrow$$ ƒ'(x) < ƒ'(2 – x)

$$\Rightarrow$$ x < 2 – x

$$\Rightarrow$$ x < 1

$$\therefore$$ $$\varphi$$(x) is decreasing on (0, 1).