If f(x)=2x+cot-1(x)+log[1+x2-x], then f(x)
Increases in [0,∞]
Decreases in [0,∞]
Neither increases nor decreases in [0,∞]
Increases in [-∞,∞]
A and D are correct.
Explanation for the correct option.
f(x)=2x+cot-1(x)+log[1+x2-x] so,
f'(x)=2-11+x2+11+x2-x×2x21+x2-1=2-11+x2+11+x2-x×x-1+x21+x2=2-11+x2-11+x2=1-11+x2+1-11+x2
Now, for all real numbers x2>0, 1+x2>1 and 1+x2>1
⇒11+x2∈(0,1]and1-11+x2∈[0,1) and 11+x2∈(0,1]⇒1-11+x2∈[0,1)
Therefore, f'x>0∀x∈ℝ, so it is increasing for all real numbers.
Hence option E is correct
Use the factor theorem to determine whether g(x) is a factor of f(x)
f(x)=22x2+5x+2;g(x)=x+2