Let b be a nonzero real number. Suppose f:ℝ→ℝ is a differentiable function such that f(0)=1 .  If the derivative f′ of f satisfies the equation f′(x)=f(x)b2+x2 for all x∈ℝ , then which of the following statements is/are TRUE?

f′(x)=f(x)b2+x2∫f′(x)f(x)dx=∫dxx2+b2⇒ln⁡|f(x)|=1btan−1⁡xb+c Now f(0)=1∴c=0∴|f(x)|=e1btan−1⁡xb⇒f(x)=±e1btan−1⁡xb since f(0)=1∴f(x)=e1btan−1⁡xbx→−xf(−x)=e−1btan−1⁡xb∴f(x)⋅f(−x)=e0=1 (option C)  and for b>0f(x)=e1btan−1⁡xb⇒f(x) is increasing for all x∈R (option A)   since f'x>0 for all x∈R ,b∈R