# Show that if 0 ≤ f' (x) ≤ 1 for all x, then the arc length of y = f (x) over [a, b] is at most √2(b − a). Show that for f (x) = x, the arc length equals √2(b − a).

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Show that if 0 ≤ f' (x) ≤ 1 for all x, then the arc length of y = f (x) over [a, b] is at most √2(b − a). Show that for f (x) = x, the arc length equals √2(b − a).

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Step 1

The arc length increases with the value of f'(x)

Means, if f'(x) increases then arc length will increase too.

So, at f'(x)=1 the arc length will be maximum

Step 2

For f'(x)=1 arc length is This is the maximum value of arc length.

Answer: If 0 ≤ f' (x) ≤ 1 for all x, then the arc length of y = f (x) over [a, b] is at most √2(b − a) [Proved]

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