Let f:ℝ→ℝ and g:ℝ→ℝ be functions satisfying fx+y=fx+fy+fxfy and fx=xgx for all x,y∈ℝ. If limx→0gx=1, then which of the following statements is/are TRUE?

(A) f is differentiable at every x∈ℝ

(B) If g0=1, then g is differentiable at everyx∈ℝ

(C) The derivative f'1 is equal to 1

(D) The derivative f'0 is equal to 1

JEE Advanced · Mathematics · 23. C&D

Let $f : ℝ \to ℝ$ and $g : ℝ \to ℝ$ be functions satisfying $f (x + y) = f (x) + f (y) + f (x) f (y) and f (x) = x g (x)$ for all $x, y \in ℝ .$ If $\lim_{x \to 0} g (x) = 1,$ then which of the following statements is/are TRUE?

A $f$ is differentiable at every $x \in ℝ$
B If $g (0) = 1,$ then $g$ is differentiable at every $x \in ℝ$
C The derivative $f^{'} (1)$ is equal to $1$
D The derivative $f^{'} (0)$ is equal to $1$

Verified Solution

Answer & Solution

Correct Answer

(A) $f$ is differentiable at every $x \in ℝ$

Step-by-step Solution

Detailed explanation

Given

f (x + y) = f (x) + f (y) + f (x) f (y)

Put

x = y = 0

in given relation.

\Rightarrow f (0) = f (0) + f (0) + f^{2} (0)

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.

Same subject

Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

Column I	Column II
(A) The minimum value of $\frac{x^2+2 x+4}{x+2}$ is	(p) 0
(B) Let $A$ and $B$ be $3 \times 3$ matrices of real numbers, where $A$ is symmetric, $B$ is skew-symmetric and $(A+B)(A-B) =$ $(A-B)(A+B)$. If $(A B)^t=(-1)^k$ $A B$, where $(A B)^t$ is the transpose of the matrix $A B$, then possible values of $k$ are	(q) 1
(C) Let $a=\log _3 \log _3 2$. An integer $k$ satisfying $1<2^{\left(-k+3^{-a}\right)}<2$, must be less than	(r) 2
(D) If $\sin \theta=\cos \phi$, then the possible values of $\frac{1}{\pi}\left(\theta \pm \phi-\frac{\pi}{2}\right)$ are	(s) 3

JEE Advanced 2008 Hard

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Column I	Column II
(A) The minimum value of \(\frac{x^2+2 x+4}{x+2}\) is	(p) 0
(B) Let \(A\) and \(B\) be \(3 \times 3\) matrices of real numbers, where \(A\) is symmetric, \(B\) is skew-symmetric and \((A+B)(A-B) =\) \((A-B)(A+B)\). If \((A B)^t=(-1)^k\) \(A B\), where \((A B)^t\) is the transpose of the matrix \(A B\), then possible values of \(k\) are	(q) 1
(C) Let \(a=\log _3 \log _3 2\). An integer \(k\) satisfying \(1<2^{\left(-k+3^{-a}\right)}<2\), must be less than	(r) 2
(D) If \(\sin \theta=\cos \phi\), then the possible values of \(\frac{1}{\pi}\left(\theta \pm \phi-\frac{\pi}{2}\right)\) are	(s) 3