Question

If f is a real-valued differentiable function satisfying |f(x) – f(y)| ≤ (x – y)² , x, y ∈ R and f(0) = 0, then f(1) equals

-1

0

2

1

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Solution

straight f apostrophe left parenthesis straight x right parenthesis space equals space limit as straight h rightwards arrow 0 of space fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction vertical line straight f apostrophe space left parenthesis straight x right parenthesis vertical line space equals space limit as straight h rightwards arrow 0 of open vertical bar fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction space close vertical bar less or equal than space limit as straight h rightwards arrow 0 of open vertical bar fraction numerator left parenthesis straight h right parenthesis squared over denominator straight h end fraction close vertical bar

⇒ |f′(x)| ≤ 0
⇒ f′(x) = 0
⇒ f(x) = constant As
f(0) = 0 ⇒ f(1) = 0

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