Question

Let f(x) = `(1 - tan x)/(4x - pi), x ne pi/4, x ∈ [0, pi/2]`. If f(x) is continuous in `[0, pi/2]`, then f`(pi/4)` is ______.

Options

• - 1

• `1/2`

• `- 1/2`

• 1

Answer 1

Let f(x) = `(1 - tan x)/(4x - pi), x ne pi/4, x ∈ [0, pi/2]`. If f(x) is continuous in `[0, pi/2]`, then f`(pi/4)` is `underline(- 1/2)`.

Explanation:

Since, f(x) is continuous in `[0, pi/2]`.

∴ It IS contmuous at x = `pi/4`.

∴ f`(pi/4) = lim_(x ->pi/4) "f"(x) = lim_(x ->pi/4) (1 - tan x)/(4x - pi)`

Applying L'Hospital rule on R.H.S., we get

f`(pi/4) = lim_(x ->pi/4) (- sec^2 x)/4`

`=> "f"(pi/4) = (- 2)/4 = (- 1)/2`