12.27

f' max at c, f''=0

1. Let \(c \in (a, b)\) and \(f''(c)\) exist. If \(f\) has any inflection point on \(x = c\)​\(\implies f''(c) = 0\)

   Proof: Let \(c\) be an inflection point, that means \(f\) is \(\cup\) in \((c - \delta, c)\) and \(f\) is \(\cap\) in \((c, c + \delta)\).

   Then \(f'\) is strictly monotonically increasing on \((c - \delta, c)\) and \(f'\) is strictly monotonically decreasing on \((c, c + \delta)\).

   Since \(f''(c) = \lim_{x \to c} \frac{f'(x) - f'(c)}{x - c}\) exists. Then \(\exists \lim_{x \to c^+} \frac{f'(x) - f'(c)}{x - c} = \lim_{x \to c^-} \frac{f'(x) - f'(c)}{x - c}\)

   \(\lim_{x \to c^+} \frac{f'(x) - f'(c)}{x - c} < 0\), \(\lim_{x \to c^-} \frac{f'(x) - f'(c)}{x - c} > 0\)​\(\implies \lim_{x \to c} \frac{f'(x) - f'(c)}{x - c} = f''(c) = 0\)

2.