Question

If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.

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Answer 1

If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = 0.

Explanation:

We have, f(x) = [x]

∴ f'(1⁺) = `lim_(x -> 1^+) ("f"(x) - "f"(1))/(x - 1) = lim_(h -> 0) ("f"(1 + "h") - "f"(1))/"h"`

`= lim_(h -> 0) ([1 + "h"] - [1])/"h"`

`=> lim_(h -> 0) (1 - 1)/"h" = 0`