Question

Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is ______.

पर्याय

• increasing on (0, 2)

• decreasing on (0, 2)

• decreasing on (0, 1) and increasing on (1, 2)

• increasing on (0, 1) and decreasing on (1, 2)

Answer 1

Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is decreasing on (0, 1) and increasing on (1, 2).

Explanation:

`phi(x)` = f(x) + f(2 – x)

`phi^'(x)` = f'(x) – f'(2 – x)

For increasing and decreasing

`phi^'(x)` = 0

⇒ `f^'(x)` = f(2 – 2)

x = 2 – x

x = 1

Interval	(0, 1)	(1, 2)
sign of `phi^'(x)`	–ve	+ve

So increasing in (1, 2) and decreasing in (0, 1)

Graph