Suppose that f^(4) is continuous and f′(c) =f′′(c) =f′′′(c) = 0, but f^(4) 6= 0. Does f(x) have a local maximum, minimum or a point of inflection at c? Justify your answer.

Answer 1

First, we need to use the following ideas...

The derivative of a function is the slope ofvthe tsngential line of the function.

This holds regardless of the function.

So, f'(c) = 0 implies the slope if the irigibal function is zero. This is the definition of a critical point (local maximum/minimum)

Secondly, f''(c) = 0, which implies the slope of the tangent line is also zero. This is the definition of an inflection point.

Lastly, we are told f'''(c) = 0, so the slope of the slope of the tangent line is also zero.

So we know that due to f"(c)= 0, it cannot be a local maximum nor a local minimum. Therefore it must be a point of inflection.