question_answer

Let \[{{S}_{1}}\] is set of minima and \[{{S}_{2}}\] is set of maxima for the curve \[y=9{{x}^{4}}+12{{x}^{3}}-36{{x}^{2}}-25,\] then:

A) \[{{S}_{1}}=\{-2,-1\};{{S}_{2}}=\{0\}\]

B) \[{{S}_{1}}=\{-2,1\};{{S}_{2}}=\{0\}\]

C) \[{{S}_{1}}=\{-2,1\};{{S}_{2}}=\{-1\}\]

D) \[{{S}_{1}}=\{-2,2\};{{S}_{2}}=\{0\}\]

Correct Answer: B

Solution :

\[y=9{{x}^{4}}+12{{x}^{3}}-36{{x}^{2}}-25\]

\[\frac{dy}{dx}=36{{x}^{3}}+36{{x}^{2}}-72x\]

\[=36x({{x}^{2}}+x-2)\]

\[=36x(x+2)(x-1).\]

\[(-2,1)\to \]points of minima

\[(0)\to \]point of maxima.