# 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)=0May need to apply first derivative test and Fermat's theorem to...

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

May need to apply first derivative test and Fermat's theorem to the fuction g=f'

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A point of inflection on a graph f(x) is where f''(x) = 0

If c is a point of inflection on the graph f(x) then, as long as f''(x) exists in the region of c then f''(c) = 0.

Fermat's Theorem says that maxima or minima of a function f(x) can be obtained by solving f'(x) = 0 where f'(x) is the gradient function of f(x).

If we differentiate f(x) once to obtain f'(x), we can find the maxima and minima of f(x) by solving f'(x) = 0. In turn, the maxima and minima of f'(x) can be obtained by solving f''(x) = 0.

So a point of inflection is where the gradient function is at a turning point.