Let f be a differentiable function from R to R such that \left|\right. f \left(\right. x \left.\right) - f \left(\right. y \left.\right) \left|\right. \leq 2 \left(\left|\right. x - y \left|\right.\right)^{3 / 2}, for all x, y \in R. If f (0) = 1 then \int_{0}^{1} f^{2} \left(\right. x \left.\right) \textrm{ } dx is equal to

Engineering

Mathematics

Monotonicity

Elementary Integrals

Concept of Derivative

Question

Let f be a differentiable function from R to R such that $| f (x) - f (y) | \leq 2 {| x - y |}^{3 / 2}$ , for all x, y $\in$ R. If f (0) = 1 then $\int_{0}^{1} f^{2} (x) dx$ is equal to

$\frac{1}{2}$

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|f(x) – f(y)| $\leq$ 2|x – y|^3/2

divide both side by |x – y|

$| \frac{f (x) - f (y)}{x - y} | \leq | 2 . | x - y |^{1 / 2}$

Apply limit x → y

|f'(y) $\leq$ 0 ⇒ f'(y) = 0 ⇒ f(y) = c ⇒ f(x) = 1|

$\int_{0}^{1} 1 . dx = 1$