Let f be a differentiable function from R to R such that |f(x) - f(y)| ≤ 2|x - y|^3/2, for all x, y ∈ R. If f(0) = 1, then ∫₁₀ f²(x) dx is equal to:<p></p>

The correct option is D
1
|f(x) - f(y)| ≤ 2|x - y|^3/2
divide both side by |x - y|
|f(x) - f(y)|/|x - y| ≤ 2|x - y|^1/2
Apply limit x→y
|f'(y)| ≤ 0
⇒f'(y) = 0
⇒f(y) = c
⇒f(x) = 1
∫₁₀ 1.dx = 1

Eduprobe

Question:

Let f be a differentiable function from R to R such that |f(x) - f(y)| ≤ 2|x - y|^3/2, for all x, y ∈ R. If f(0) = 1, then ∫₁₀ f²(x) dx is equal to:

Solution: