`L e tf(x)={x^3+x^2+3x+sinx|(3+s in1/x ,),x!=0. 0x=0` then the number of point where `f(x)` attains its minimum value is_____

Answer

1

Explanation

`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