Continuous Calculus

Calculus Level 4

True or False?

Suppose \(f\) is a real-valued continuous function defined on \([0, 1]\) such that \((2x − 1)(2f(x) − x) > 0\) for all \(x \neq \frac 12.\)

If the first derivative \(f'\left(\frac 12 \right)\) exists, then \(f'\left(\frac 12 \right)\) cannot be smaller than \(\frac 12\).

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