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Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Derivative of Function.

Let f(x)=x[x] where [x] denotes the greatest integer smaller than or equal to x where x is not an integer, what is f'(x)?

- 2x
- [x]
- 2[x]
- it does not exist.

Equation

Derivative

Algebra

But try the problem first...

Answer: [x].

Source

Suggested Reading

B.Stat Objective Problem 756

Challenges and Thrills of Pre-College Mathematics by University Press

First hint

\(f(x)=x[x]\) where f(x) discontinuous at integer points so non differentiable at those pts

so we find for non-integer points

Second Hint

if x is not an integer then [x] is a constant and not equal to x

Final Step

x is non integer so here [x] is treated as constant here . Let [x] = a then f(x) =ax => f'(x)= a = [x] . Hence option [x] is correct.

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