Prove that if (c, f(c)) is a point of inflection of the graph of f and f'' exists in an open interval that contains c, then f''(c)=0 Hint: Apply the First Derivative Test and Fermat's Theorem to the function g=f'

G=f′ is differentiable on an open interval containing c. Since (c,f(c))
is a point of inflection, the concavity
changes at x=c

Therefore, f′′(x) changes signs at x=c.

Thus, by the First Derivative Test, f′ has a local extremum at x=c

Hence, by Fermat's Theorem f′′(c)=0

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