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Let \(f:[-2a,2a]\to \mathbb{R}\) be an odd function. If the left-hand derivative at \(x=a\) is zero and \(f(x)=f(2a-x)\) for all \(x\in(a,2a)\), what is the value of the left-hand derivative at \(x=-a\)?

1. a
2. b
3. a
4. does not exist

Correct answer: a

Solution

From \(f(x)=f(2a-x)\), differentiating gives \(f'(x)=-f'(2a-x)\) where applicable. Setting \(x=a\) relates the derivatives at \(a\) and \(a\), and using the odd nature of \(f\) transfers the result to \(-a\). The intended answer is \(a\).

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