Let b be a nonzero real number. Suppose f:ℝ→ℝ is a differentiable function such that f0=1. If the derivative f^(') of f satisfies the equation…

Question

Let b be a nonzero real number. Suppose f:ℝ→ℝ is a differentiable function such that f0=1. If the derivative f^(') of f satisfies the equation f^(')x=fx/(b^2+x^2) for all x∈ℝ, then which of the following statements is/are TRUE?

MARKS Content Team · Accepted Answer

Correct answer: (A) If b>0, then f is an increasing function; (C) fx f-x=1 for all x∈ℝ. f^(')x=fx/(b^2+x^2)⇒(f^(')x)/fx=1/(b^2+x^2) ⇒lnfx=1/btan^(-1)x/b+C ∴f0=1 ∴0=0+C⇒C=0 ∴ lnfx=1/btan^(-1)x/b ⇒fx=e^(1btan-1)x/b ∴f^(')x=e^(1btan-1)x/b.1/(b^2+x^2)>0 for x∈R ⇒fx is increasing for x∈R. fx.f-x=e^(1btan-1)x/b·e^(-1btan-1)x/b=e^0=1.

Let $b$ be a nonzero real number. Suppose $f : ℝ \to ℝ$ is a differentiable function such that $f (0) = 1$ . If the derivative $f^{'}$ of $f$ satisfies the equation $f^{'} (x) = \frac{f (x)}{b^{2} + x^{2}}$ for all $x \in ℝ,$ then which of the following statements is/are TRUE?

Solution

Relevant Questions

Let b be a nonzero real number. Suppose f:ℝ→ℝ is a differentiable function such that f0=1. If the derivative f' of f satisfies the equation f'x=fxb2+x2 for all x∈ℝ, then which of the following statements is/are TRUE?

Solution

Relevant Questions

Let $b$ be a nonzero real number. Suppose $f : ℝ \to ℝ$ is a differentiable function such that $f (0) = 1$ . If the derivative $f^{'}$ of $f$ satisfies the equation $f^{'} (x) = \frac{f (x)}{b^{2} + x^{2}}$ for all $x \in ℝ,$ then which of the following statements is/are TRUE?