A) The cubic has minima at both \[\sqrt{\frac{p}{3}}\] and \[-\sqrt{\frac{p}{3}}\]
B) The cubic has maxima at both \[\sqrt{\frac{p}{3}}\]and \[-\sqrt{\frac{p}{3}}\]
C) The cubic has minima at \[\sqrt{\frac{p}{3}}\] and maxima at \[-\sqrt{\frac{p}{3}}\]
D) The cubic has minima at \[-\sqrt{\frac{p}{3}}\] and maxima at \[\sqrt{\frac{p}{3}}\]
Correct Answer: C
Solution :
Let \[f\left( x \right)={{x}^{3}}-px+q\] \[f'\left( x \right)=3{{x}^{2}}-p\] For maxima or minima \[f'\left( x \right)=0\,\,\,\Rightarrow x\pm \sqrt{\frac{p}{3}}\] \[f''\left( x \right)=6x\Rightarrow f''\left( x \right)>0\] for \[x=\sqrt{\frac{p}{3}}\] and \[f''\left( x \right)<0\] for \[x=-\sqrt{\frac{p}{3}}\]
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