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Question Number 37143 by rahul 19 last updated on 09/Jun/18

Why are following statements wrong?  a) There exists a function with domain   R satisfying f(x)<0 ∀x , f′(x)>0∀x and  f′′(x)>0∀x.    b) If f′′(c)=0 then (c,f(c)) is an inflection  point.

$$\mathrm{Why}\:\mathrm{are}\:\mathrm{following}\:\mathrm{statements}\:\mathrm{wrong}? \\ $$ $$\left.\mathrm{a}\right)\:\mathrm{There}\:\mathrm{exists}\:\mathrm{a}\:\mathrm{function}\:\mathrm{with}\:\mathrm{domain}\: \\ $$ $$\mathrm{R}\:\mathrm{satisfying}\:\mathrm{f}\left(\mathrm{x}\right)<\mathrm{0}\:\forall\mathrm{x}\:,\:\mathrm{f}'\left(\mathrm{x}\right)>\mathrm{0}\forall\mathrm{x}\:\mathrm{and} \\ $$ $$\mathrm{f}''\left(\mathrm{x}\right)>\mathrm{0}\forall\mathrm{x}. \\ $$ $$ \\ $$ $$\left.\mathrm{b}\right)\:\mathrm{If}\:\mathrm{f}''\left(\mathrm{c}\right)=\mathrm{0}\:\mathrm{then}\:\left(\mathrm{c},\mathrm{f}\left(\mathrm{c}\right)\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{inflection} \\ $$ $$\mathrm{point}. \\ $$

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