# If $S_{1}\; and\; S_{2}$ are respectively the sets of local minimum and local maximum points of the function,$f(x)=9x^{4}+12x^{3}-36x^{2}+25,x\: \epsilon \: \mathbb{R},$ then :   Option 1) $S_{1}=\left \{ -2 \right \};S_{2}=\left \{ 0,1 \right \}$ Option 2) $S_{1}=\left \{ -2 ,1\right \};S_{2}=\left \{ 0 \right \}$ Option 3) $S_{1}=\left \{ -2 ,0\right \};S_{2}=\left \{ 1 \right \}$   Option 4) $S_{1}=\left \{ -1\right \};S_{2}=\left \{ 0,2 \right \}$

$f(x)=9x^{4}+12x^{3}-36x^{2}+25\; \; \; \; \; x\: \epsilon \; \mathbb{R}$

$y=9x^{4}+12x^{3}-36x^{2}+25$

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

$=36(x^{3}+x^{2}-2)$

$=36x(x^{3}+x-2)$

$=36x(x+2)(x-1)$

$\frac{-\; \; \; +\; \; \; -\; \; \; +}{-2\; \; \; 0\; \; \; +1}$

$S_{1}=point\; of\; minima=\left \{ -2,1 \right \}$

$S_{2}=point\; of\; mixima=\left \{ 0 \right \}$

Option 1)

$S_{1}=\left \{ -2 \right \};S_{2}=\left \{ 0,1 \right \}$

Option 2)

$S_{1}=\left \{ -2 ,1\right \};S_{2}=\left \{ 0 \right \}$

Option 3)

$S_{1}=\left \{ -2 ,0\right \};S_{2}=\left \{ 1 \right \}$

Option 4)

$S_{1}=\left \{ -1\right \};S_{2}=\left \{ 0,2 \right \}$

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