Condider that $f''$ has a double root, what can be concluded of $f$? I mean is there any clue of how it acts on that particular point? Does it always happen at maximum or minimum?
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Consider, say, $f(x)=x^3$. Then $f'(x)=3x^2$ has a double root at $0$ but $0$ is neither a max nor a min for $f(x)$. – lulu Mar 23 '20 at 16:19
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1well,if you think about Rolle's theorm, if $f''$ has $2$ roots, then $f'$ itself must have 3 and $f$ , $4$ – sai-kartik Mar 23 '20 at 16:20
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Not always, Let f"=12x^2 so f =x^4 +cx+d, as you can see f" has a double root in x=0, but f has a minimum or maximum in x=0 if c=0
PeXa
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The fact that the root is repeated is not so important as the fact that $f''$ vanishes there. In that case, provided $f''(x+\mathrm dx)$ changes sign according as $\mathrm dx\ne 0$ changes sign across the vanishing point, then we conclude that there is a point of inflexion of $f$ there -- that is, a swap in concavity.
Allawonder
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