JEE Advanced · Mathematics · 29. Differential Eqns

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?

A If $b > 0,$ then $f$ is an increasing function
B If $b < 0,$ then $f$ is a decreasing function
C $f (x) f (- x) = 1$ for all $x \in ℝ$
D $f (x) - f (- x) = 0$ for all $x \in ℝ$

Verified Solution

Answer & Solution

Correct Answer

(C) $f (x) f (- x) = 1$ for all $x \in ℝ$

Step-by-step Solution

Detailed explanation

f^{'} (x) = \frac{f (x)}{b^{2} + x^{2}} \Rightarrow \frac{f^{'} (x)}{f (x)} = \frac{1}{b^{2} + x^{2}}

\Rightarrow l n f (x) = \frac{1}{b} \tan^{- 1} \frac{x}{b} + C

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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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?

Step-by-step Solution

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?