Testing Second Derivatives Using Limits

00:00 5 marks

Using the epsilon-delta definition of the limit, prove that if \(f”(c)\) exists and is positive, then the function \(f\) has a local minimum at the point \(c\). In your answer, outline the key steps that connect the limit formulation of \(f”(c)\) to the conclusion about the curvature of \(f\) around \(c\).

Hint Hint

Hint

Recall the epsilon-delta definition of the second derivative and relate a positive value of \(f”(c)\) to the increasing nature of \(f’\) near \(c\).

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Answer Answer

Complete Answer

By applying the epsilon-delta definition to the second derivative, one shows that if \(f”(c)>0\), then for small enough \(h\), the difference quotient remains positive, indicating that \(f’\) is increasing near \(c\). This implies that \(f\) changes from decreasing to increasing at \(c\), hence a local minimum exists at \(c\).

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Answer Explanation Answer Explanation

Common Errors Common Errors

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