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I'm studying second derivative tests, concavity and inflection points in khan academy https://www.khanacademy.org/math/differential-calculus/derivative_applications/concavity-inflection-points/v/concavity-concave-upwards-and-concave-downwards-intervals but Salman takes na arbitrary function that could be $x^{10}$, but the first derivative is always a parabola, and the second derivative is always a line. I think I'm missing something fundamental here. Could somebody explain why I'm thinking wrong?

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I just skimmed the video. I think you are mistaking something. $f(x)$ is not arbitrary. In his example, $f(x)$ is a cubic, in which case we would expect $f'(x)$ to be quadratic and $f''(x)$ to be a line.

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