Question

Let f(x) = tan^–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.

Options

• increasing in `(0, π/2)`

• decreasing in `(0, π/2)`

• increasing in `(0, π/4)` and decreasing in `(π/4, π/2)`

• decreasing in `(0, π/4)` and increasing in `(π/4, π/2)`

Answer 1

Let f(x) = tan^–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is `underlinebb("increasing in" (0, π/2))`.

Explanation:

f(x) = tan^–1{`phi`(x)}

`phi`(x) is increasing function for 0 < x < `π/2` is `phi`'(x) > 0 for 0 < x < `π/2`

We have f'(x) = `1/(1 + phi^2(x)) xx phi^'(x)`

⇒ f'(x) > 0 for 0 < x < `π/2`

∴ f(x) is increasing for `∈(0, π/2)`