Question

Let \( f : (a,b) \to \mathbb{R} \) be a differentiable function on \((a,b)\). Which of the following statements is/are true?

• A. \(f'(x)>0\) in \((a,b)\) implies that \(f\) is increasing in \((a,b).\)
• B. \(f\) is increasing in \((a,b)\) implies that \(f'>0\) in \((a,b).\)
• C. If \(f'(x_0)>0\) for some \(x_0 \in (a,b)\), then there exists a \(\delta>0\) such that \(f(x)>f(x_0)\) for all \(x \in (x_0, x_0 + \delta).\)
• D. If \(f'(x_0)>0\) for some \(x_0 \in (a,b)\), then \(f\) is increasing in a neighbourhood of \(x_0.\)

Hint:

A positive derivative implies local monotonicity, but the converse is not always true. Check counterexamples like \(f(x) = x^3.\)

Answer 1

The Correct Option is A, C

Step 1: Statement (A).
By the Mean Value Theorem, if \(f'(x)>0\) everywhere, then for \(x_2>x_1\), \[ f(x_2) - f(x_1) = f'(c)(x_2 - x_1)>0, \] so \(f\) is strictly increasing. Hence (A) is true.
Step 2: Statement (B).
If \(f\) is increasing, \(f'\) need not be positive everywhere (e.g. \(f(x) = x^3\) near 0). Hence (B) is false.
Step 3: Statements (C) and (D).
If \(f'(x_0)>0\), then by definition of derivative, \(f\) increases locally near \(x_0\). Thus, both (C) and (D) are true.
Step 4: Conclusion.
Hence, correct statements are (A), (C), and (D).