1.

Let f(x)={{:(|x^(3)+x^(2)+3x+sinx(3+sin(1)/(x)), x ne 0), (0,x=0):} Then the number of points where f(X) attians its minimum value is _______.

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SOLUTION :`f(x)={{:(x^(3)+x^(2)+3x+sinx3+sin(1/x)),(xne0):}`
Let `g(x)=x^(3)+x^(2)+3x+sinx`
`therefore g(x)=3x^(2)+2x+3+cosx`
`=3(x^(2)+(2x)/(3)+1)+cosx`
`=3{(x+1/3)^(2)+8/9}+coxgt0`
and `2lt3+sin(1/x)lt4`
Hence minimum value of `f(x) is 0 at x=0`
Hence number of POINTS =1


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