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Black Advanced Problems in Mathematics

Exam: Black Advanced Problems in Mathematics
Subject: Mathematics
Chapter: Indefinite and Definite Integration

Black Advanced Problems in Mathematics · Mathematics · Indefinite and Definite Integration

Let f:(0, 1) arrow (0, 1) be a differentiable function such that...

Let

f:(0, 1) \rightarrow (0, 1)

be a differentiable function such that

f'(x) \neq 0

for all

x \in (0, 1)

and

f\left(\frac{1}{2}\right) = \frac{\sqrt{3}}{2}

. Suppose for all

x

,

\lim_{t \to x} \frac{\int_0^t \sqrt{1 - (f(s))^2} ds - \int_0^x \sqrt{1 - (f(s))^2} ds}{f(t) - f(x)} = f(x)

. Then the value of

f \left(\frac{1}{4}\right)belongs to :

Options

A\left\{ \frac{\sqrt{7}}{4}, \frac{\sqrt{15}}{4} \right\}
B\left\{ \frac{\sqrt{7}}{3}, \frac{\sqrt{15}}{3} \right\}
C\left\{ \frac{\sqrt{7}}{2}, \frac{\sqrt{15}}{2} \right\}
D\left\{ \sqrt{7}, \sqrt{15} \right\}

Correct answer

A. \left\{ \frac{\sqrt{7}}{4}, \frac{\sqrt{15}}{4} \right\}

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