Question

Let f(x) be a non-negative continuous function such that the area bounded by the curve y = f(x), X-axis and the ordinates x = `pi/4` and x = `beta > pi/4` is `(beta sin beta + pi/4 cos beta + sqrt(2)beta)`. Then `"f"(pi/2)` is ______.

Options

• `1 - pi/4 + sqrt(2)`

• `1 - pi/4 - sqrt(2)`

• `pi/4 - sqrt(2) + 1`

• `pi/4 + sqrt(2) - 1`

Answer 1

Let f(x) be a non-negative continuous function such that the area bounded by the curve y = f(x), X-axis and the ordinates x = `pi/4` and x = `beta > pi/4` is `(beta sin beta + pi/4 cos beta + sqrt(2)beta)`. Then `"f"(pi/2)` is `1 - pi/4 + sqrt(2)`.

Explanation:

According to the given condition,

`int_(pi/4)^beta "f"(x)"d"x = beta sin beta + pi/4 cos beta + sqrt(2)beta`

Differentiating w.r.t. `beta`, we get

`"f"(beta) = sin beta + beta cos beta - pi/4 sin beta + sqrt(2)`

∴ `"f"(pi/2) = sin  pi/2 + pi/2 cos  pi/2 - pi/4 + sqrt(2)`

= `1 - pi/4 + sqrt(2)`